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MATH.-STAT. 


SIM  ISAAC  MIBWf  OM« 


NEWTON'S  PRINCIPIA. 


THE 


MATHEMATICAL  PRINCIPLES 


OF 


NATURAL   PHILOSOPHY, 

BY   SIR   ISAAC  NEWTON; 

TRANSLATED  INTO  ENGLISH  BY  ANDREW  MOTTE. 


TO  WHICH   IS  ADDKTV 


NEWTON'S  SYSTEM  OF  THE  WORLD ; 

With  a  Portrait  taken  from  the  Bust  in  the  Royal  Observatory  at  Greenwich. 

FIRST  AMERICAN  EDITION,  CAREFULLY  REVISED  AND   CORRECTED, 

WITH  A  LIFE  OF  THE  AUTHOR,  BY  PI.  W.  CHITTENDEN,  M.  A.,  &e. 


NEW-YORK 

PUBLISHED  BY  DANIEL  ADEE,   45   LIBERTY   STREET. 


p*- 


Kntered  according  to  Act  of  Congress,  in  the  year  1846,  by 

DANIEL   ADEE. 
3!Ltht  Clerk's  Office  ut'tiie  Southern  Oisli:ct  Court  of  New-York. 


TWuey  *  Lockwoof,  Stom 
16  Spruce  St.  N.  Y. 


DEDICATION. 


TO  THE 

TEACHERS  OF  THE  NORMAL  SCHOOL 

OF  THE  STATE  OF  NEW-YORK. 

GENTLEMEN  ! 

A  stirring  freshness  in  the  air,  and  ruddy  streaks  upon  the 
horizon  of  the  moral  world  betoken  the  grateful  dawning  of  a  new 
ora.  The  days  of  a  drivelling  instruction  are  departing.  With 
us  is  the  opening  promise  of  a  better  time,  wherein  genuine  man 
hood  doing  its  noblest  work  shall  have  adequate  reward. 
TEACHER  is  the  highest  and  most  responsible  office  man  can  fill. 
Its  dignity  is,  and  will  yet  be  held  commensurate  with  its  duty — 
a  duty  boundless  as  man's  intellectual  capacity,  and  great  as  his 
moral  need — a  duty  from  the  performance  of  which  shall  emanate 
an  influence  not  limited  to  the  now  and  the  here,  but  which  surely 
will,  as  time  flows  into  eternity  and  space  into  infinity,  roll  up,  a 
measureless  curse  or  a  measureless  blessing,  in  inconceivable 
swellings  along  the  infinite  curve.  It  is  an  office  that  should  be 
esteemed  of  even  sacred  import  in  this  country.  Ere  long  a  hun 
dred  millions,  extending  from  the  Atlantic  to  the  Pacific,  from 
Baffin's  Bay  to  that  of  Panama,  shall  call  themselves  American 
citizens.  What  a  field  for  those  two  master-passions  of  the  hu 
man  soul — the  love  of  Rule,  and  the  love  of  Gain !  How  shall 
our  liberties  continue  to  be  preserved  from  the  graspings  of  Am 
bition  and  the  corruptions  of  Gold  ?  Not  by  Bills  of  Rights 


4  DEDICATION. 

Constitutions,  and  Statute  Books  ;  but  alone  by  the  rightly  culti 
vated  hearts  and  heads  of  the  PEOPLE.  They  must  themselves 
guard  the  Ark.  It  is  yours  to  tit  them  for  the  consecrated 
charge.  Look  well  to  it :  for  you  appear  clothed  in  the  majesty 
of  great  power  !  It  is  yours  to  fashion,  and  to  inform ,  to  save, 
and  to  perpetuate.  You  are  the  Educators  of  the  People  :  you 
are  the  prime  Conservators  of  the  public  weal.  Betray  your 
trust,  and  the  sacred  fires  would  go  out,  and  the  altars  crumble 
into  dust :  knowledge  become  lost  in  tradition,  and  Christian  no 
bleness  a  fable !  As  you,  therefore,  are  multiplied  in  number, 
elevated  in  consideration,  increased  in  means,  and  fulfill,  well  and 
faithfully,  all  the  requirements  of  true  Teachers,  so  shall  our  fa 
voured  land  lift  up  her  head  among  the  nations  of  the  earth,  and 
call  herself  blessed. 

In  conclusion,  Gentlemen,  to  you,  as  the  conspicuous  leaders 
in  the  vast  and  honourable  labour  of  Educational  Helbrm,  ana 
Popular  Teaching,  the  First  American  Edition  of  the  PRINCIPIA  ol 
Newton — the  greatest  wrork  of  the  greatest  Teacher — is  most 
respectfully  dedicated. 

N.  W.  CHITTENDEN. 


INTRODUCTION  TO  THE  AMERICAN  EDITION. 


THAT  the  PRINCIPIA  of  Newton  should  have  remained  so  gen 
erally  unknown  in  this  country  to  the  present  day  is  a  somewhat 
remarkable  fact ;  because  the  name  of  the  author,  learned  with 
the  very  elements  of  science,  is  revered  at  every  hearth-stone 
where  knowledge  and  virtue  are  of  chief  esteem,  while,  abroad, 
in  all  the  high  places  of  the  land,  the  character  which  that  name 
recalls  is  held  up  as  the  noblest  illustration  of  what  MAN  may  be, 
and  may  do,  in  the  possession  and  manifestation  of  pre-eminent 
intellectual  and  moral  worth  ;  because  the  work  is  celebrated,  not 
only  in  the  history  of  one  career  and  one  mind,  but  in  the  history 
of  all  achievement  and  human  reason  itself;  because  of  the  spirit 
of  inquiry,  which  has  been  aroused,  and  which,  in  pursuing  its 
searchings,  is  not  always  satisfied  with  stopping  short  of  the  foun 
tain-head  of  any  given  truth  ;  and,  finally,  because  of  the  earnest 
endeavour  that  has  been  and  is  constantly  going  on,  in  many 
sections  of  the  Republic,  to  elevate  the  popular  standard  of  edu 
cation  and  give  to  scientific  and  other  efforts  a  higher  and  a 
better  aim. 

True,  the  PRINCIPIA  has  been  hitherto  inaccessible  to  popular 
use.  A  few  copies  in  Latin,  and  occasionally  one  in  English  may 
be  found  in  some  of  our  larger  libraries,  or  in  the  possession  of 
some  ardent  disciple  of  the  great  Master.  But  a  d^ad  language 
in  the  one  case,  and  an  enormous  price  in  both,  particularly  in 
that  of  the  English  edition,  have  thus  far  opposed  very  sufficient 
obstacles  to  the  wide  circulation  of  the  work.  It  is  now,  how 
ever,  placed  within  the  reach  of  all.  And  in  performing  this  la 
bour,  the  utmost  care  has  been  taken,  by  collation,  revision,  and 
otherwise,  to  render  the  First  American  Edition  the  most  accurate 
and  beautiful  in  our  language.  u  Le  plus  beau  monument  que 
l?  on  puisse  clever  a  la  gloire  de  Newton,  c'est  une  bonne  edition 
de  ses  ouvrages :"  and  a  monument  like  unto  that  we  would  here 


V:  INTRODUCTION    TO 

set  up.  The  PRINCIPIA,  above  all,  glows  with  the  immortality  of 
a  transcendant  mind.  Marble  and  brass  dissolve  and  pass  away  ; 
but  the  true  creations  of  genius  endure,  in  time  and  beyond  time, 
forever  :  high  upon  the  adamant  of  the  indestructible,  they  send 
forth  afar  and  near,  over  the  troublous  waters  of  life,  a  pure,  un 
wavering,  quenchless  light  whereby  the  myriad  myriads  of  barques, 
richly  laden  with  reason,  intelligence  and  various  faculty,  are 
guided  through  the  night  and  the  storm,  by  the  beetling  shore 
and  the  hidden  rock,  the  breaker  and  the  shoal,  safely  into  havens 
calm  and  secure. 

To  the  teacher  and  the  taught,  the  scholar  and  the  student,  the 
devotee  of  Science  and  the  worshipper  of  Truth,  the  PRINCIPIA 
must  ever  continue  to  be  of  inestimable  value.  If  to  educate 
means,  not  so  much  to  store  the  memory  with  symbols  and  facts, 
as  to  bring  forth  the  faculties  of  the  soul  and  develope  them  to  the 
full  by  healthy  nurture  and  a  hardy  discipline,  then,  what  so  effec 
tive  to  the  accomplishment  of  that  end  as  the  study  of  Geometri 
cal  Synthesis  ?  The  Calculus,  in  some  shape  or  other,  is,  indeed, 
necessary  to  the  successful  prosecution  of  researches  in  the  higher 
branches  of  philosophy.  But  has  not  the  Analytical  encroached 
upon  the  Synthetical,  and  Algorithmic  Formulae  been  employed 
when  not  requisite,  either  for  the  evolution  of  truth,  or  even  its 
apter  illustration  ?  To  each  method  belongs,  undoubtedly,  an 
appropriate  use.  Newton,  himself  the  inventor  of  Fluxions, 
censured  the  handling  of  Geometrical  subjects  by  Algebraical 
calculations  ;  and  the  maturest  opinions  which  he  expressed  were 
additionally  in  favour  of  the  Geometrical  Method.  His  prefer 
ence,  so  strongly  marked,  is  not  to  be  reckoned  a  mere  matter  oi 
taste  ;  and  his  authority  should  bear  with  preponderating  weight 
upon  the  decision  of  every  instructor  in  adopting  what  may  be 
deemed  the  best  plan  to  insure  the  completes!  mental  develop 
ment.  Geometry,  the  vigorous  product  of  remote  time  ;  blended 
with  the  earliest  aspirations  of  Science  and  the  earliest  applica 
tions  of  Art ;  as  well  in  the  measures  of  music  as  in  the  move 
ment  of  spheres  ;  as  wholly  in  the  structure  of  the  atom  as  in  that 
of  the  world;  directing  MOTION  and  shaping  APPEARANCE;  in  a 
wonl,  *t  the  moulding  of  the  created  all,  is,  in  comprehensive 


THE    AMERICAN    EDITION.  Vll 

view,  the  outward  form  of  that  Inner  Harmony  of  which  and  in 
which  all  things  are.  Plainly,  therefore,  this  noble  study  has 
other  and  infinitely  higher  uses  than  to  increase  the  power  of  ab 
straction.  A  more  general  and  thorough  cultivation  of  it  should 
oe  strenuously  insisted  on.  Passing  from  the  pages  of  Euclid  or 
Legendre,  might  not  the  student  be  led,  at  the  suitable  time,  to 
those  of  the  PRINCIPIA  wherein  Geometry  may  be  found  in  varied 
use  from  the  familiar  to  the  sublime  ?  The  profoundest  and  the 
happiest  results,  it  is  believed,  would  attend  upon  this  enlargement 
of  our  Educational  System. 

Let  the  PRINCIPIA,  then,  be  gladly  welcomed  into  every  Hall 
where  a  TRUE  TEACHER  presides.  And  they  who  are  guided  to 
the  diligent  study  of  this  incomparable  work,  who  become 
strengthened  by  its  reason,  assured  by  its  evidence,  and  enlight 
ened  by  its  truths,  and  who  rise  into  loving  communion  with  the 
great  and  pure  spirit  of  its  author,  will  go  forth  from  the  scenes 
of  their  pupilage,  and  take  their  places  in  the  world  as  strong- 
minded,  right-hearted  men — such  men  as  the  Theory  of  our 
Government  contemplates  and  its  practical  operation  absolutely 
demands. 


LIFE  OF 

SIE  ISAAC  NEWTON. 


Nec  fas  est  proprius  mortal?  attingere  Divos. — HALLEY. 


FROM  the  thick  darkness  of  the  middle  ages  man's  struggling 
spirit  emerged  as  in  new  birth  ;  breaking  out  of  the  iron  control 
of  that  period  ;  growing  strong  and  confident  in  the  tug  and  din 
of  succeeding  conflict  and  revolution,  it  bounded  forwards  and 
upwards  with  resistless  vigour  to  the  investigation  of  physical  and 
moral  truth ;  ascending  height  after  height ;  sweeping  afar  over 
the  earth,  penetrating  afar  up  into  the  heavens  ;  increasing  in  en 
deavour,  enlarging  in  endowment ;  every  where  boldly,  earnestly 
out-stretching,  till,  in  the  AUTHOR  of  the  PRINCIPIA,  one  arose, 
who,  grasping  the  master-key  of  the  universe  and  treading  its 
celestial  paths,  opened  up  to  the  human  intellect  the  stupendous 
realities  of  the  material  world,  and,  in  the  unrolling  of  its  harmo 
nies,  gave  to  the  human  heart  a  new  song  to  the  goodness,  wis 
dom,  and  majesty  of  the  all-creating,  all-sustaining,  all-perfect 
God. 

Sir  Isaac  Newton,  in  whom  the  rising  intellect  seemed  to  attain, 
as  it  were,  to  its  culminating  point,  was  born  on  the  25th  of  De 
cember,  O.  S.  1642 — Christmas  day — at  Woolsthorpe,  in  the 
parish  of  Colsterworth,  in  Lincolnshire.  His  father,  John  New 
ton,  died  at  the  age  of  thirty-six,  and  only  a  few  months  after  his 
marriage  to  Harriet  Ayscough,  daughter  of  James  Ayscough,  oi 
Rutlandshire.  Mrs.  Newton,  probably  wrought  upon  by  the 
early  loss  of  her  husband,  gave  premature  birth  to  her  only  and 
posthumous  child,  of  which,  too,  from  its  extreme  diminutiveness, 
she  appeared  likely  to  be  soon  bereft.  Happily,  it  was  otherwise 
decreed !  The  tiny  infant,  on  whose  little  lips  the  breath  of  life 


10  LIFE    OF    SIR    ISAAC    NEWTON. 

so  doubtingly  hovered,  lived  ; — lived  to  a  vigorous  maturity,  to  a 
hale  old  age  ; — lived  to  become  the  boast  of  his  country,  the  won 
der  of  his  time,  and  the  "ornament  of  his  srjecies." 

Beyond  the  grandfather,  Robert  Newton,  the  descent  of  Sir 
Isaac  cannot  with  certainty  be  traced.  Two  traditions  were  held 
in  the  family :  one,  that  they  were  of  Scotch  extraction  ;  the 
other,  that  they  came  originally  from  Newton,  in  Lancashire, 
dwelling,  for  a  time,  however,  at  Westby,  county  of  Lincoln,  be 
fore  the  removal  to  and  purchase  of  Woolsthorpe — about  a  hundred 
years  before  this  memorable  birth. 

The  widow  Newton  was  left  with  the  simple  means  of  a  com 
fortable  subsistence.     The  Woolsthorpe  estate  together  with 
small  one  which  she  possessed  at  Sewstern,  in  Leicestershire,  yield 
ed  her  an  income  of  some  eighty  pounds ;  and  upon  this  limited  sum, 
she  had  to  rely  chiefly  for  the  support  of  herself,  and  the  educa 
tion  of  her  child.     She   continued  his  nurture  for  three  years, 
when,  marrying  again,  she  confided  the  tender  charge  to  the  care 
of  her  own  mother. 

Great  genius  is  seldom  marked  by  precocious  development ; 
and  young  Isaac,  sent,  at  the  usual  age,  to  two  day  schools  at 
Skillington  and  Stoke,  exhibited  no  unusual  traits  of  character. 
In  his  twelfth  year,  he  was  placed  at  the  public  school  at  Gran- 
tham,  and  boarded  at  the  house  of  Mr.  Clark,  an  apothecary. 
But  even  in  this  excellent  seminary,  his  mental  acquisitions  con 
tinued  for  a  while  unpromising  enough  :  study  apparentlv  had  no 
charms  for  him ;  he  was  very  inattentive,  and  ranked  low  in  the 
school.  One  day,  however,  the  boy  immediately  above  our  seem 
ingly  dull  student  gave  him  a  severe  kick  in  the  stomach  ;  Isaac, 
deeply  affected,  but  with  no  outburst  of  passion,  betook  himself, 
with  quiet,  incessant  toil,  to  his  books  ;  he  quickly  passed  above 
the  offending  classmate  ;  yet  there  he  stopped  not ;  the  strong 
spirit  was,  for  once  and  forever,  awakened,  and,  yielding  to  itb 
noble  impulse,  he  speedily  took  up  his  position  at  the  head  of  all. 

His  peculiar  character  began  now  rapidly  to  unfold  itself. 
Close  application  grew  to  be  habitual.  Observation  alternated 
with  reflection.  "  A  sober,  silent,  thinking  lad,"  yet,  the  wisest 
and  the  kindliest,  the  indisputable  leader  of  his  fellows.  Gener- 


LIFE    OF    SIR    ISA  VC    NEWTON.  11 

osity,  modesty,  and  a  love  of  truth  distinguished  him  then  as  ever 
afterwards.  He  did  not  often  join  his  classmates  in  play  ;  but  he 
would  contrive  for  them  various  amusements  of  a  scientific  kind. 
Paper  kites  he  introduced ;  carefully  determining  their  best  form 
and  proportions,  and  the  position  and  number  of  points  whereby 
to  attach  the  string.  He  also  invented  paper  lanterns  ;  these 
served  ordinarily  to  guide  the  way  to  school  in  winter  mornings, 
but  occasionally  for  quite  another  purpose  ;  they  were  attached  to 
the  tails  of  kites  in  a  dark  night,  to  the  dismay  of  the  country  people 
dreading  portentous  comets,  and  to  the  immeasureable  delight  ol 
his  companions.  To  him,  however,  young  as  he  was,  life  seemed 
to  have  become  an  earnest  thing.  When  not  occupied  with  his 
studies,  his  mind  would  be  engrossed  with  mechanical  contrivances ; 
now  imitating,  now  inventing.  He  became  singularly  skilful  in  the 
use  of  his  little  saws,  hatchets,  hammers,  and  other  tools.  A 
windmill  was  erected  near  Grantham  ;  during  the  operations  ol 
the  workmen,  he  was  frequently  present ;  in  a  short  time,  he  had 
completed  a  perfect  working  model  of  it,  which  elicited  general 
admiration.  Not  content,  however,  with  this  exact  imitation,  he 
conceived  the  idea  of  employing,  in  the  place  of  sails,  animal  power  , 
and,  adapting  the  construction  of  his  mill  accordingly,  he  enclosed 
in  it  a  mouse,  called  the  miller,  and  which  by  acting  on  a  sort  ot 
treadvvheel,  gave  motion  to  the  machine.  He  invented,  too,  a 
mechanical  carriage — having  four  wheels,  and  put  in  motion  with 
a  handle  worked  by  the  person  sitting  inside.  The  measurement 
of  time  early  drew  his  attention.  He  h'rst  constructed  a  water 
clock,  in  proportions  somewhat  like  an  old-fashioned  house  clock. 
The  index  of  the  dial  plate  was  turned  by  a  piece  of  wood  acted 
upon  by  dropping  water.  This  instrument,  though  long  used  by 
himself,  and  by  Mr.  Clark's  family,  did  not  satisfy  his  inquiring 
mind.  His  thoughts  rose  to  the  sun  ;  and,  by  careful  and  oft-re 
peated  observations  of  the  solar  movements,  he  subsequently 
formed  many  dials.  One  of  these,  named  Isaac's  dial,  was  the 
accurate  result  of  years'  labour,  and  was  frequently  referred  to 
for  the  hour  of  the  day  by  the  country  people. 

May  we  not  discern  in  these  continual  efforts — the  diligent  re 
search^  the  patient  meditation,  the  aspiring  glance,  and  the  energy 


12  LIFE    OF    SIR    ISAAC    NEWTON. 

of  discovery — the  stirring  elements  of  that  wondrous  spirit, 
which,  clear,  calm,  and  great,  moved,  in  after  years,  through 
deep  onward  through  deep  of  Nature's  mysteries,  unlocking  her 
strongholds,  dispelling  darkness,  educing  order — everywhere  si 
lently  conquering. 

Newton  had  an  early  and  decided  taste  for  drawing.  Pictures, 
taken  sometimes  from  copies,  but  often  from  life,  and  drawn, 
coloured  and  framed  by  himself,  ornamented  his  apartment.  He 
was  skilled  also,  in  poetical  composition,  "  excelled  in  making 
verses  ;"  some  of  these  were  borne  in  remembrance  and  repeated, 
seventy  years  afterward,  by  Mrs.  Vincent,  for  whom,  in  early 
youth,  as  Miss  Storey,  he  formed  an  ardent  attachment.  She 
was  the  sister  of  a  physician  resident  near  Woolsthorpe ;  but 
Newton's  intimate  acquaintance  with  her  began  at  Grantham. 
where  they  were  both  numbered  among  the  inmates  of  the  same 
house.  Two  or  three  years  younger  than  himself,  of  great  per 
sonal  beauty,  and  unusual  talent,  her  society  afforded  him  the 
greatest  pleasure  ;  and  their  youthful  friendship,  it  is  believed, 
gradually  rose  to  a  higher  passion ;  but  inadequacy  of  fortune 
prevented  their  union.  Miss  Storey  was  afterwards  twice  mar 
ried  ;  Newton,  never;  his  esteem  for  her  continued  unabated 
during  life,  accompanied  by  numerous  acts  of  attention  and 
kindness. 

In  1656,  Newton's  mother  was  again  left  a  widowr,  and  took 
up  her  abode  once  more  at  Woolsthorpe.  He  was  now  fifteen 
years  of  age,  and  had  made  great  progress  in  his  studies  ;  but  she, 
desirous  of  his  help,  and  from  motives  of  economy,  recalled  him 
from  school.  Business  occupations,  however,  and  the  manage 
ment  of  the  farm,  proved  utterly  distasteful  to  him.  When  sent  to 
Grantham  Market  on  Saturdays,  he  would  betake  himself  to  his 
former  lodgings  in  the  apothecary's  garret,  where  some  of  Mr. 
Clark's  old  books  employed  his  thoughts  till  the  aged  and  trust 
worthy  servant  had  executed  the  family  commissions  and  announced 
the  necessity  of  return  :  or,  at  other  times,  our  young  philosopher 
would  seat  himself  under  a  hedge,  by  the  wayside,  and  continue 
his  studies  till  the  same  faithful  personage — proceeding  alone  to 
the  town  and  completing  the  day's  business — stopped  as  he  re- 


LIFE    OF    SIR    ISAAC    NEWTON,  13 

turned.  The  more  immediate  affairs  of  the  farm  received  no 
better  attention.  In  fact,  his  passion  for  study  grew  daily  more 
absorbing,  and  his  dislike  for  every  other  occupation  more  in 
tense.  His  mother,  therefore,  wisely  resolved  to  give  him  all  the 
advantages  which  an  education  could  confer.  He  was  sent  back 
to  Grantham  school,  where  he  remained  for  some  months  in  busy 
preparation  for  his  academical  studies.  At  the  recommendation 
of  one  of  his  uncles,  who  had  himself  studied  at  Trinity  College, 
Cambridge,  Newton  proceeded  thither,  and  was  duly  admitted. 
on  the  5th  day  of  June  1660,  in  the  eighteenth  year  of  his  age. 

The  eager  student  had  now  entered  upon  a  new  and  wider 
field ;  and  we  find  him  devoting  himself  to  the  pursuit  of  know 
ledge  with  amazing  ardour  and  perseverance.  Among  other  sub 
jects,  his  attention  was  soon  drawn  to  that  of  Judicial  Astrology 
He  exposed  the  folly  of  this  pseudo-science  by  erecting  a  figure 
with  the  aid  of  one  or  two  of  the  problems  of  Euclid  ; — and  thus 
began  his  study  of  the  Mathematics.  His  researches  into  this 
science  were  prosecuted  with  unparallelled  vigour  and  success. 
Regarding  the  propositions  contained  in  Euclid  as  self-evident 
truths,  he  passed  rapidly  over  this  ancient  system — a  step  which 
he  afterward  much  regretted — and  mastered,  without  further  pre 
paratory  study,  the  Analytical  Geometry  of  Descartes.  Wallis's 
Arithmetic  of  Infinites,  Saunderson's  Logic,  and  the  Optics  of 
Kepler,  he  also  studied  with  great  care  ;  writing  upon  them 
many  comments ;  and,  in  these  notes  on  Wallis's  work  was  un 
doubtedly  the  germ  of  his  fluxionary  calculus.  His  progress  was 
so  great  that  he  found  himself  more  profoundly  versed  than  his  tutor 
in  many  branches  of  learning.  Yet  his  acquisitions  were  not 
gotten  with  the  rapidity  of  intuition ;  but  they  were  thoroughly 
made  and  firmly  secured.  Quickness  of  apprehension,  or  Intel 
lectual  nimbleness  did  not  belong  to  him.  He  saw  too  far  :  his, 
insight  was  too  deep.  He  dwelt  fully,  cautiously  upon  the  least 
subject ;  while  to  the  consideration  of  the  greatest,  he  brought  a 
massive  strength  joined  with  a  matchless  clearness,  that,  regard 
less  of  the  merely  trivial  or  unimportant,  bore  with  unerring  sa 
gacity  upon  the  prominences  of  the  subject,  and,  grappling  with 
its  difficulties,  rarely  failed  to  surmount  them. 


14  LIFE    OF    SIR    ISAAC    NEWTON 

His  early  and  fast  friend,  Dr.  Barrow — in  compass  of  inven 
tion  only  inferior  to  Newton — who  had  been  elected  Professor 
of  Greek  in  the  University,  in  1660,  was  made  Lucasian  Profes 
sor  of  Mathematics  in  1663,  and  soon  afterward  delivered  his 
Optical  Lectures  :  the  manuscripts  of  these  were  revised  by  New 
ton,  and  several  oversights  corrected,  and  many  important  sug 
gestions  made  by  him  ;  but  they  were  not  published  till  1669. 

In  the  year  1665,  he  received  the  degree  of  Bachelor  of  Arts ; 
and,  in  1666,  he  entered  upon  those  brilliant  and  imposing  dis 
coveries  which  have  conferred  inappreciable  benefits  upon  science, 
and  immortality  upon  his  own  name. 

Newton,  himself,  states  that  he  was  in  possession  of  his  Method 
of  Fluxions,  "  in  the  year  1666,  or  before."  Infinite  quantities 
had  long  been  a  subject  of  profound  investigation ;  among  the 
ancients  by  Archimedes,  and  Pappus  of  Alexandria ;  among  the 
moderns  by  Kepler,  Cavaleri,  Roberval,  Fermat  and  Wallis. 
With  consummate  ability  Dr.  Wallis  had  improved  upon  the  la- 
hours  of  his  predecessors :  with  a  higher  power,  Newton  moved 
forwards  from  where  Wallis  stopped.  Our  author  first  invented 
his  celebrated  BINOMIAL  THEOREM.  And  then,  applying  this 
Theorem  to  the  rectification  of  curves,  and  to  the  determination 
of  the  surfaces  and  contents  of  solids,  and  the  position  of  their 
centres  of  gravity,  he  discovered  the  general  principle  of  deducing 
the  areas  of  curves  from  the  ordinate,  by  considering  the  area  as 
a  nascent  quantity,  increasing  by  continual  fluxion  in  the  propor 
tion  of  the  length  of  the  ordinate,  and  supposing  the  abscissa 
to  increase  uniformly  in  proportion  to  the  time.  Regarding  lines 
as  generated  by  the  motion  of  points,  surfaces  by  the  motion  of 
lines,  and  solids  by  the  motion  of  surfaces,  and  considering  that 
the  ordinates,  abscissae,  &c.,  of  curves  thus  formed,  vary  accord 
ing  to  a  regular  law  depending  on  the  equation  of  the  curve, 
he  deduced  from  this  equation  the  velocities  with  which  these 
quantities  are  generated,  and  obtained  by  the  rules  of  infinite 
series,  the  ultimate  value  required.  To  the  velocities  with  which 
every  line  or  quantity  is  generated,  he  gave  the  name  of  FLUX 
IONS,  and  to  the  lines  or  quantities  themselves,  that  of  FLUENTS. 
A  discovery  that  successively  baffled  the  acutest  and  strongest 


LIFE    OF    SIR    ISAAC    NEWTON.  15 

intellects  : — that,  variously  modified,  has  proved  of  incalculable 
service  in  aiding  to  develope  the  most  abstruse  and  the  highest 
'ruths  in  Mathematics  and  Astronomy :  and  that  was  of  itself 
enough  to  render  any  name  illustrious  in  the  crowded  Annals  of 
Science. 

At  this  period,  the  most  distinguished  philosophers  were  direct 
ing  all  their  energies  to  the  subject  of  light  and  the  improvement 
of  the  refracting  telescope.  Newton,  having  applied  himself  to 
the  grinding  of  "optic  glasses  of  other  figures  than  spherical,"  ex 
perienced  the  impracticability  of  executing  such  lenses  ;  and  con 
jectured  that  their  defects,  and  consequently  those  of  refracting 
telescopes,  might  arise  from  some  other  cause  than  the  imperfect 
convergency  of  rays  to  a  single  point.  He  accordingly  "procured 
a  triangular  glass  prism  to  try  therewith  the  celebrated  phenom 
ena  of  colours."  His  experiments,  entered  upon  with  zeal,  and 
conducted  with  that  industry,  accuracy,  and  patient  thought,  for 
which  he  was  so  remarkable,  resulted  in  the  grand  conclusion, 
that  LIGHT  WAS  NOT  HOMOGENEOUS,  BUT  CONSISTED  OF  RAYS, 

SOME  OF  WHICH  WERE  MORE    REFRANGIBLE    THAN    OTHERS.       This 

profound  and  beautiful  discovery  opened  up  a  new  era  in  the 
History  of  Optics.  As  bearing,  however,  directly  upon  the  construc 
tion  of  telescopes,  he  saw  that  a  lens  refracting  exactly  like  a  prism 
would  necessarily  bring  the  different  rays  to  different  foci,  at 
different  distances  from  the  glass,  confusing  and  rendering  the 
vision  indistinct.  Taking  for  granted  that  all  bodies  produced 
spectra  of  ^  jtial  length,  he  dismissed  all  further  consideration  of 
the  refracting  instrument,  and  took  up  the  principle  of  reflection. 
Rays  of  all  colours,  he  found,  were  reflected  regularly,  so  that  the 
angle  of  reflection  was  equal  to  the  angle  of  incidence,  and  hence 
he  concluded  that  ojitical  instruments  might  be  brought  to  any 
degree  of  perfection  imaginable,  provided  reflecting  specula  of 
the  requisite  figure  and  finish  could  be  obtained.  At  this  stage 
of  his  optical  researches,  he  was  forced  to  leave  Cambridge  on 
account  of  the  plague  which  was  then  desolating  England. 

He  retired  to  Woolsthorpe.  The  old  manor-house,  in  which  he 
was  born,  was  situated  in  a  beautiful  little  valley,  on  the  west  side 
of  the  river  Witham  ;  and  here  in  the  quiet  home  of  his  boyhood, 

2 


16  LIFE    OF    SIR    ISAAC    NEWTON. 

he  passed  his  days  in  serene  contemplation,  while  the  stalking 
pestilence  was  hurrying  its  tens  of  thousands  into  undistinguisha  • 
ble  graves. 

Towards  the  close  of  a  pleasant  day  in  the  early  autumn  of 
1666,  he  was  seated  alone  beneath  a  tree,  in  his  garden,  absorbed 
in  meditation.  He  was  a  slight  young  man  ;  in  the  twenty-fourth 
year  of  his  age  ;  his  countenance  mild  and  full  of  thought.  For 
a  century  previous,  the  science  of  Astronomy  had  advanced  with 
rapid  strides.  The  human  mind  had  risen  from  the  gloom  and 
bondage  of  the  middle  ages,  in  unparalleled  vigour,  to  unfold  the 
system,  to  investigate  the  phenomena,  and  to  establish  the  laws 
of  the  heavenly  bodies.  Copernicus,  Tycho  Brahe,  Kepler, 
Galileo,  and  others  had  prepared  and  lighted  the  way  for  him 
who  was  ta  give  to  their  labour  its  just  value,  and  to  their  genius 
its  true  lustre.  At  his  bidding  isolated  facts  were  to  take  order 
as  parts  of  one  harmonious  whole,  and  sagacious  conjectures  grow 
luminous  in  the  certain  splendour  of  demonstrated  truth.  And 
this  ablest  man  had  come — was  here.  His  mind,  familiar  with 
the  knowledge  of  past  effort,  and  its  unequalled  faculties  develop 
ed  in  transcendant  strength,  was  now  moving  on  to  the  very 
threshold  of  Its  grandest  achievement.  Step  by  step  the  untrod 
den  path  was  measured,  till,  at  length,  the  entrance  seemed  dis 
closed,  and  the  tireless  explorer  to  stand  amid  the  first  opening 
wonders  of  the  universe. 

The  nature  of  gravity — that  mysterious  power  which  causes 
all  bodies  to  descend  towards  the  centre  of  the  earth — had,  in 
deed,  dawned  upon  him.  And  reason  busily  united  link  to  link 
of  that  chain  which  was  yet  to  be  traced  joining  the  least  to  the 
vastest,  the  most  remote  to  the  nearest,  in  one  harmonious  bond. 
From  the  bottoms  of  the  deepest  caverns  to  the  summits  of  the 
highest  mountains,  this  power  suffers  no  sensible  change  :  may  not 
its  action,  then,  extend  to  the  moon  ?  Undoubtedly  :  and  furthei 
reflection  convinced  him  that  such  a  power  might  be  .sufficient  for 
retaining  that  luminary  in  her  orbit  round  the  earth.  But,  though 
this  power  suffers  no  sensible  variation,  in  the  little  change  of 
distance  from  the  earth's  centre,  at  which  we  may  place  our- 
.'«lves,  yet,  at  the  distance  of  the  moon,  :miy  not  its  force  undergo 


LIFE    OF    SIR    ISAAC    NEWTON.  17 

more  or  less  diminution  ?  The  conjecture  appeared  most  proba 
ble  :  and,  in  order  to  estimate  what  the  degree  of  diminution 
might  be,  he  considered  that  if  the  moon  be  retained  in  her  orbit 
by  the  force  of  gravity,  the  primary  planets  must  also  be  carried 
round  the  sun  by  the  like  power;  and,  by  comparing  the  periods 
of  the  several  planets  with  their  distances  from  the  sun,  he  found 
that,  if  they  were  held  in  their  courses  by  any  power  like  gravity, 
its  strength  must  decrease  in  the  duplicate  proportion  of  the  in 
crease  of  distance.  In  forming  this  conclusion,  he  supposed  the 
planets  to  move  in  perfect  circles,  concentric  to  the  sun.  Now 
was  this  the  law  of  the  moon's  motion  ?  Was  such  a  force,  em 
anating  from  the  earth  and  directed  to  the  moon,  sufficient,  when 
diminished  as  the  square  of  the  distance,  to  retain  her  in  her 
orbit  ?  To  ascertain  this  master-fact,  he  compared  the  space 
through  which  heavy  bodies  fall,  in  a  second  of  time,  at  a  given 
distance  from  the  centre  of  the  earth,  namely,  at  its  surface,  with 
the  space  through  which  the  moon  falls,  as  it  were,  to  the  earth, 
in  the  same  time,  while  revolving  in  a  circular  orbit.  He  was 
absent  from  books  ;  and,  therefore,  adopted,  in  computing  the 
earth's  diameter,  the  common  estimate  of  sixty  miles  to  a  degree 
of  latitude  as  then  in  use  among  geographers  and  navigators. 
The  result  of  his  calculations  did  not,  ot  course,  answer  his  ex 
pectations  ;  hence,  he  concluded  that  some  other  cause,  beyond  the 
reach  of  observation — analogous,  perhaps,  to  the  vortices  of  Des 
cartes — joined  its  action  to  that  of  the  power  of  gravity  upon  the 
rnooil.  Though  by  no  means  satisfied,  he  yet  abandoned  awhile 
further  inquiry,  and  remained  totally  silent  upon  the  subject. 

These  rapid  marches  in  the  career  of  discovery,  combined  with 
the  youth  of  Newton,  seem  to  evince  a  penetration  the  most 
lively,  and  an  invention  the  most  exuberant.  But  in  him  there 
was  a  conjunction  of  influences  as  extraordinary  as  fortunate. 
Study,  unbroken,  persevering  and  profound  carried  on  its  inform 
ing  and  disciplining  work  upon  a  genius,  natively  the  greatest, 
and  rendered  freest  in  its  movements,  and  clearest  in  its  vision, 
through  the  untrammelling  and  enlig}  tenirig  power  of  religion. 
And,  in  this  happy  concurrence,  are  to  be  sought  the  elements  of 
those  amazing  abilities,  which,  grasping,  with  equal  facility,  the 


18  LIFE    OF    SIR    ISAAC    NEWTON. 

minute  and  the  stupendous,  brought  these  successively  to  light, 
and  caused  science  to  make  them  her  own. 

In  1667,  Newton  was  made  a  Junior  Fellow  ;  and,  in  the  year 
following,  he  took  his  degree  of  Master  of  Arts,  and  was  appoint 
ed  to  a  Senior  Fellowship. 

On  his  return  to  Cambridge,  in  1668,  he  resumed  his  optical 
labours.  Having  thought  of  a  delicate  method  of  polishing  metal, 
he  proceeded  to  the  construction  of  his  newly  projected  reflect 
ing  telescope  ;  a  small  specimen  of  which  he  actually  made  with 
his  own  hands,  It  was  six  inches  long  ;  and  magnified  about 
forty  times  ; — a  power  greater  than  a  refracting  instrument  of  six 
feet  tube  could  exert  with  distinctness.  Jupiter,  with  his  four 
satellites,  and  the  horns,  or  moon-like  phases  of  Venus  were 
plainly  visible  through  it.  THIS  WAS  THE  FIRST  REFLECTING 

TELESCOPE     EVER     EXECUTED     AND     DIRECTED     TO     THE     HEAVENS. 

He  gave  an  account  of  it,  in  a  letter  to  a  friend,  dated  February  23d, 
1668-9 — a  letter  which  is  also  remarkable  for  containing  the  firs' 
allusion  to  his  discoveries  "  concerning  the  nature  of  light."  En 
couraged  by  the  success  of  his  first  experiment,  he  again  executed 
with  his  own  hands,  not  long  afterward,  a  second  and  superior 
instrument  of  the  same  kind.  The  existence  of  this  having  come 
to  the  knowledge  of  the  Royal  Society  of  London,  in  1671,  they 
requested  it  of  Newton  for  examination.  He  accordingly  sent  it 
to  them,  It  excited  great  admiration;  it  was  shown  to  the  king* 
a  drawing  and  description  of  it  was  sent  to  Paris ;  and  the  tele- 
scope  itself  was  carefully  preserved  in  the  Library  of  the  Society. 
Newton  lived  to  see  his  invention  in  public  use,  and  of  eminent 
service  in  the  cause  of  science. 

In  the  spring  of  1669,  he  wrote  to  his  friend  Francis  Aston, 
Esq.,  then  about  setting  out  on  his  travels,  a  letter  of  advice  and 
directions,  it  was  dated  May  18th,  and  is  interesting  as  exhibit 
ing  some  of  the  prominent  features  in  Newton's  character. 
Thus  : — 

"  Since  in  your  letter  you  give  me  so  much  liberty  of  spending 
my  judgment  about  what  may  be  to  your  advantage  in  travelling, 
1  shall  do  it  more  freely  than  perhaps  otherwise  would  have  been 
decent,  Fir,ct,  then,  I  will  lay  down  some  general  rules,  most  of 


LIFE    OF    SIR    ISAAC    NEWTON.  19 

which,  I  bolieA  e,  you  have  considered  already ;  but  if  any  of 
them  be  new  to  you,  they  may  excuse  the  rest  ;  if  none  at  all, 
yet  is  my  punishment  more  in  writing  than  yours  in  reading. 

"When  you  come  into  any  fresh  company.  1.  Observe  their 
humours.  2.  Suit  your  own  carriage  thereto,  by  which  insinua 
tion  you  will  make  their  converse  more  free  and  open.  3.  Let 
your  discourse  be  more  in  queries  and  doubtings  than  peremptory 
assertions  or  disputings,  it  being  the  design  of  travellers  to  learn, 
not  to  teach.  Besides,  it  will  persuade  your  acquaintance  that 
you  have  the  greater  esteem  of  them,  and  so  make  them  more 
ready  to  communicate  what  they  know  to  you  ;  whereas  nothing 
sooner  occasions  disrespect  and  quarrels  than  peremptoriness. 
You  will  find  little  or  no  advantage  in  seeming  wiser  or  much 
more  ignorant  than  your  company.  4.  Seldom  discommend  any 
thing  though  never  so  bad,  or  do  it  but  moderately,  lest  you  be 
unexpectedly  forced  to  an  unhandsome  retraction.  It  is  safer  to 
commend  any  thing  more  than  it  deserves,  than  to  discommend 
a  thing  so  much  as  it  deserves;  for  commendations  meet  not 
so  often  with  oppositions,  or,  at  least,  are  not  usually  so  ill  re 
sented  by  men  that  think  otherwise,  as  discommendations ;  and 
you  will  insinuate  into  men's  favour  by  nothing  sooner  than  seem 
ing  to  approve  and  commend  what  they  like  ;  but  beware  o 
doing  it  by  comparison.  5.  If  you  be  affronted,  it  is  better,  in  c 
foreign  country,  to  pass  it  by  in  silence,  and  with  a  jest,  though 
with  some  dishonour,  than  to  endeavour  revenge ;  for,  in  the  first 
case,  your  credit's  ne'er  the  worse  when  you  return  into  England, 
or  come  into  other  company  that  have  not  heard  of  the  quarrel. 
But,  in  the  second  case,  you  may  bear  the  marks  of  the  quarrel 
while  you  live,  if  you  outlive  it  at  all.  But,  if  you  find  yoursell 
unavoidably  engaged,  'tis  best,  I  think,  if  you  can  command  your 
passion  and  language,  to  keep  them  pretty  evenly  at  some  certain 
moderate  pitch,  not  much  heightening  them  to  exasperate  your 
adversary,  or  provoke  his  friends,  nor  letting  them  grow  overmuch 
dejected  to  make  him  insult.  In  a  word,  if  you  can  keep  reason 
above  passion,  that  and  watchfulness  will  be  your  best  defendants. 
To  which  purpose  you  may  consider,  that,  though  such  excuses 
is  this — He  provok't  me  so  much  I  could  not  forbear — may  pass 


20  LIFE    OF    SIR    ISAAC    NEWTON. 

among  friends,  yet   amongst  strangers  they  are  insignificant,  ina 
only  argue  a  traveller's  weakness. 

"  To  these  I  may  add  some  general  heads  for  inquiries  or  ob 
servations,  such  as  at  present  I  can  think  on.  As,  1.  To  observe 
the  policies,  wealth,  and  state  affairs  of  nations,  so  far  as  a  soli- 
fary  traveller  may  conveniently  do.  2.  Their  impositions  upon 
all  sorts  of  people,  trades,  or  commodities,  that  are  remarkable. 
3.  Their  laws  and  customs,  how  far  they  differ  from  ours.  4. 
Their  trades  and  arts  wherein  they  excel  or  come  short  of  us  in 
England.  5.  Such  fortifications  as  you  shall  meet  with,  their 
fashion,  strength,  and  advantages  for  defence,  and  other  such  mili 
tary  affairs  as  are  considerable.  6.  The  power  and  respect  be 
longing  to  their  degrees  of  nobility  or  magistracy.  7.  It  will  not 
be  time  misspent  to  make  a  catalogue  of  the  names  and  excellen 
cies  of  those  men  that  are  most  wise,  learned,  or  esteemed  in  any 
nation.  8.  Observe  the  mechanism  and  manner  of  guiding  ships. 

9.  Observe  the  products  of  Nature  in  several  places,  especially  in 
mines,  with  the  circumstances  of  mining  and  of  extracting  metals 
or  minerals   out  of  their  ore,  and  of  refining  them  ;  and  if  you 
meet  with   any  transmutations   out    of  their  own    species    into 
another  (as  out  of  iron  into  copper,  out  of  any  metal  into  quick 
silver,  out  of  one  salt  into  another,  or  into  an  insipid  body,  &c.), 
those,  above  all,  will  be  worth  your  noting,  being  the  most  lucif- 
erous,  and  many  times  lucriferous  experiments,  too,  in  philosophy. 

10.  The   prices   of  diet   and  other  things.      11.  And  the  staple 
commodities  of  places. 

"  These  generals  (such  as  at  present  I  could  think  of),  if  they 
will  serve  for  nothing  else,  yet  they  may  assist  you  in  drawing  up 
a  model  to  regulate  your  travels  by.  '  As  for  particulars,  these  that 
follow  are  all  that  1  can  now  think  of,  viz. ;  whether  at  Schem- 
nitium,  in  Hungary  (where  there  are  mines  of  gold,  copper,  iron, 
vitriol,  antimony,  &c.).  they  change  iron  into  copper  by  dissolving 
»t  in  a  vitriolate  water,  which  they  find  in  cavities  of  rocks  in  the 
mines,  and  then  melting  the  slimy  solution  in  a  stroi  ig  fire,  which 
in  the  cooling  proves  copper.  The  like  is  said  to  be  done  in  other 
places,  which  I  cannot  now  remember ;  perhaps,  too,  it  may  be 
lone  in  Italy.  For  about  twenty  or  thirty  years  agone  there  was 


LIFE    OF    SIR    ISAAC    NEWTON,  21 

a  certain  vitriol  came  from  thence  (called  Roman  vitriol),  but  of 
a  nobler  virtue  than  that  which  is  now  called  by  that  name  ; 
which  vitriol  is  not  now  to  be  gotten,  because,  perhaps,  they  make 
a  greater  gain  by  some  such  trick  as  turning  iron  into  copper 
with  it  than  by  selling  it.  2.  Whether,  in  Hungary,  Sclavonia, 
Bohemia,  near  the  town  Eila,  or  at  the  mountains  of  Bohemia 
near  Silesia,  there  be  rivers  whose  waters  are  impregnated  with 
gold  ;  perhaps,  the  gold  being  dissolved  by  some  corrosive  water 
like  aqua  regis,  and  the  solution  carried  along  with  the  stream, 
that  runs  through  the  mines.  And  whether  the  practice  of  laying 
mercury  in  the  rivers,  till  it  be  tinged  with  gold,  and  then  strain 
ing  the  mercury  through  leather,  that  the  gold  may  stay  behind, 
be  a  secret  yet,  or  openly  practised.  3.  There  is  newly  con 
trived,  in  Holland,  a  mill  to  grind  glasses  plane  withal,  and  I 
think  polishing  them  too  ;  perhaps  it  will  be  worth  the  while  to  see 

it.     4.  There  is  in  Holland  one Borry,  who  some  years  since 

was  imprisoned  by  the  Pope,  to  have  extorted  from  him  secrets 
(as  I  am  told)  of  great  worth,  both  as  to  medicine  and  profit,  but 
he  escaped  into  Holland,  where  they  have  granted  him  a  guard. 
I  think  he  usually  goes  clothed  in  green.  Pray  inquire  what  you 
can  of  him,  and  whether  his  ingenuity  be  any  profit  to  the  Dutch. 
You  may  inform  yourself  whether  the  Dutch  have  any  tricks  to 
keep  their  ships  from  being  all  worm-eaten  in  their  voyages  to 
the  Indies.  Whether  pendulum  clocks  do  any  service  in  finding 
out  the  longitude,  &c. 

"  I  am  very  weary,  and  shall  not  stay  to  part  with  a  long 
compliment,  only  I  wish  you  a  good  journey,  and  God  be  with 
you." 

It  was  not  till  the  month  of  June,  1669,  that  our  author  made 
known  his  Method  of  Fluxions.  He  then  communicated  the 
work  which  he  had  composed  upon  the  subject,  and  entitled, 
ANALYSIS  PER  EQUATIONES  NUMERO  TERMINORUM  INFINITAS, 
to  his  friend  Dr.  Barrow.  The  latter,  in  a  letter  dated  20th  of  the 
same  month,  mentioned  it  to  Mr.  Collins,  and  transmitted  it  to 
him,  on  the  31st  of  July  thereafter.  Mr.  Collins  greatly  approv> 
ed  of  the  work ;  took  a  copy  of  it ;  and  sent  the  original  back 
to  Dr.  Barrow.  During  the  same  and  the  two  following  years,  Mr 


<  LIFE    OF    SIR    ISAAC    NEWTON. 

Collins,  by  his  extensive  correspondence,  spread  the  knowledge 
of  this  discovery  among  the  mathematicians  in  England,  Scotland, 
France,  Holland  and  Italy. 

Dr.  Barrow,  having  resolved  to  devote  himself  to  Theology, 
resigned  the  Lucasian  Professorship  of  Mathematics,  in  1669,  in 
favour  of  Newton,  who  accordingly  received  the  appointment  to 
the  vacant  chair. 

During  the  years  1669,  1670,  and  1671,  our  author,  as  such 
Professor,  delivered  a  course  of  Optical  Lectures.  Though  these 
contained  his  principal  discoveries  relative  to  the  different  re- 
frangibility  of  light,  yet  the  discoveries  themselves  did  not  be 
come  publicly  known,  it  seems,  till  he  communicated  them  to  the 
Royal  Society,  a  few  weeks  after  being  elected  a  member  there 
of,  in  the  spring  of  1671-2.  He  now  rose  rapidly  in  reputation, 
and  was  soon  regarded  as  foremost  among  the  philosophers  of  the 
age.  His  paper  on  light  excited  the  deepest  interest  in  the  Royal 
Society,  who  manifested  an  anxious  solicitude  to  secure  the  author 
from  the  "  arrogations  of  others,"  and  proposed  to  publish  his 
discourse  in  the  monthly  numbers  in  which  the  Transactions  were 
given  to  the  world.  Newton,  gratefully  sensible  of  these  expres 
sions  of  esteem,  willingly  accepted  of  the  proposal  for  publication. 
He  gave  them  also,  at  this  time,  the  results  of  some  further  ex 
periments  in  the  decomposition  and  re-composition  of  light : — that 
the  same  degree  of  refrangibility  always  belonged  to  the  same 
colour,  and  the  same  colour  to  the  same  degree  of  refrangibility : 
that  the  seven  different  colours  of  the  spectrum  were  original,  or 
simple,  and  that  whiteness^  or  white  light  was  a  compound  of  all 
these  seven  colours. 

The  publication  of  his  new  doctrines  on  light  soon  called  forth 
violent  opposition  as  to  their  soundness.  Hooke  and  Huygens — 
men  eminent  for  ability  and  learning — were  the  most  conspicuous 
of  the  assailants.  And  though  Newton  effectually  silenced  all  his 
adversaries,  yet  he  felt  the  triumph  of  little  gain  in  comparison 
.vith  the  loss  his  tranquillity  had  sustained.  He  subsequently  re- 
narked  in  allusion  to  this  controversy — and  to  one  with  whom 
he  was  destined  to  have  a  longer  and  a  bitterer  conflict — "  I  was 
so  persecuted  with  discussions  arising  from  the  publication  of  m  v 


LIFE    OF    SIR    ISAAC    NEWTON.  23 

theory  ot  light,  that  I  blamed  my  own  imprudence  for  parting 
with  so  substantial  a  blessing  as  rny  quiet  to  run  after  a  shadow.7' 

In  a  communication  to  Mr.  Oldenburg,  Secretary  of  the  Royal 
Society,  in  1672,  our  author  stated  many  valuable  suggestions  re 
lative  to  the  construction  of  REFLECTING  MICROSCOPES  which  he 
considered  even  more  capable  of  improvement  than  telescopes. 
He  also  contemplated,  about  the  same  time,  an  edition  of  Kirick- 
huysen's  Algebra,  with  notes  and  additions;  partially  arranging, 
as  an  introduction  to  the  work,  a  treatise,  entitled,  A  Method  of 
Fluxions ;  but  he  finally  abandoned  the  design.  This  treatise, 
however,  he  resolved,  or  rather  consented,  at  a  late  period  of  his 
life,  to  put  forth  separately  ;  and  the  plan  would  probably  have 
been  carried  into  execution  had  riot  his  death  intervened.  It  was 
translated  into  English,  and  published  in  1736  by  John  Colson, 
Professor  of  Mathematics  in  Cambridge. 

Newton,  it  is  thought,  made  his  discoveries  concerning  the 
INFLECTION  and  DIFFRACTION  of  light  before  1674.  The  phe 
nomena  of  the  inflection  of  light  had  been  first  discovered  more 
than  ten  years  before  by  Grimaldi.  And  Newton  began  by  re 
peating  one  of  the  experiments  of  the  learned  Jesuit — admitting 
a  beam  of  the  sun's  light  through  a  small  pin  hole  into  a  dark 
chamber :  the  light  diverged  from  the  aperture  in  the  form  of  a, 
cone,  and  the  shadows  of  all  bodies  placed  in  this  light  were 
larger  than  might  have  been  expected,  and  surrounded  with  three 
coloured  fringes,  the  nearest  being  widest,  and  the  most  remote 
the  narrowest.  Newton,  advancing  upon  this  experiment,  took 
exact  measures  of  the  diameter  of  the  shadow  of  a  human  hair, 
and  of  the  breadth  of  the  fringes,  at  different  distances  behind  it, 
and  discovered  that  these  diameters  and  breadths  were  not  pro 
portional  to  the  distances  at  which  they  were  measured.  He 
hence  supposed  that  the  rays  which  passed  by  the  edge  of  the 
hair  were  deflected  or  turned  aside  from  it,  as  if  by  a  repulsive 
force,  the  nearest  rays  suffering  the  greatest,  the  more  remote  a 
less  degree  of  deflection.  In  explanation  of  the  coloured  fringes, 
he  queried  :  whether  the  rays  which  differ  in  refrangibility  do  not 
differ  also  in  flexibility,  and  whether  they  are  n<»t,  by  these  dif 
ferent  inflections,  separated  from  one  another,  so  as  after  separa- 


<  LIFE    OF    SIR    ISAAC    NEWTON. 

tion  to  make  the  colours  in  the  three  fringes  above  described  ? 
Also,  whether  the  rays,  in  passing  by  the  edges  and  sides  ol 
bodies,  are  not  bent  several  times  backwards  and  forwards  with 
an  eel-like  motion — the  three  fringes  arising  from  three  such 
bendings  ?  His  inquiries  on  this  subject  were  here  interrupted 
and  fiever  renewed. 

His  Theory  of  the  COLOURS  of  NATURAL  BODIES  was  commu 
nicated  to  the  Royal  Society,  in  February,  1675.  This  is  justly 
regarded  as  one  of  the  profoundest  of  his  speculations.  The  fun 
damental  principles  of  the  Theory  in  brief,  are : — That  bodies 
possessing  the  greatest  refractive  powers  reflect  the  greatest 
quantity  of  light ;  and  that,  at  the  confines  of  equally  refracting 
media,  there  is  no  reflection.  That  the  minutest  particles  of  al 
most  all  natural  bodies  are  in  some  degree  transparent.  That 
between  the  particles  of  bodies  there  are  pores,  or  spaces,  either 
empty  or  filled  with  media  of  a  less  density  than  the  particles 
themselves.  That  these  particles,  and  pores  or  spaces,  have  some 
definite  size.  Hence  he  deduced  the  Transparency,  Opacity,  and 
colours  of  natural  bodies.  Transparency  arises  from  the  particles 
and  their  pores  being  too  small  to  cause  reflection  at  their  com 
mon  surfaces — the  light  all  passing  through  ;  Opacity  from  the 
opposite  cause  of  the  particles  and  their  pores  being  sufficiently 
large  to  reflect  the  light  which  is  "  stopped  or  stifled7'  by  the 
multitude  of  reflections ;  and  colours  from  the  particles,  accord 
ing  to  their  several  sizes,  reflecting  rays  of  one  colour  and  trans 
mitting  those  of  another — or  in  other  words,  the  colour  that 
meets  the  eye  is  the  colour  reflected,  while  all  the  other  rays  are 
transmitted  or  absorbed. 

Analogous  in  origin  to  the  colours  of  natural  bodies,  he  con 
sidered  the  COLOURS  OF  THIN  PLATES.  This  subject  was  interest 
ing  and  important,  and  had  attracted  considerable  investigation. 
He,  however,  was  the  first  to  determine  the  law  of  the  produc 
tion  of  these  colours,  arid,  during  the  same  year  made  known  the 
results  of  his  researches  herein  to  the  Royal  Society.  His  mode 
of  procedure  in  these  experiments  was  simple  and  curious.  He 
placed  a  double  convex  lens  of  a  large  known  radius  of  curvature, 
the  flat  surface  of  a  plano-convex  object  glass.  Thus,  from 


UFE    OF    SIR    ISAAC    NEWTON.  25 

their  point  of  contact  at  the  centre,  to  the  circumference  of  the 
lens,  he  obtained  plates  of  air,  or  spaces  varying  from  the  ex- 
tremest  possible  thinness,  by  slow  degrees,  to  a  considerable  thick 
ness.  Letting  the  light  fall,  every  different  thickness  of  this 
plate  of  air  gave  different  colours — the  point  of  contact  of  the 
lens  and  glass  forming  the  centre  of  numerous  concentric  colored 
nags.  Now  the  radius  of  curvature  of  the  lens  being  known,  the 
thickness  of  the  plate  of  air,  at  any  given  point,  or  where  any  par 
ticular  colour  appeared,  could  be  exactly  determined.  Carefully 
noting,  therefore,  the  order  in  which  the  different  colours  ap 
peared,  he  measured,  with  the  nicest  accuracy,  the  different  thick* 
nesses  at  which  the  most  luminous  parts  of  the  rings  were  pro 
duced,  whether  the  medium  were  air,  water,  or  mica — all  these 
substances  giving  the  same  colours  at  different  thicknesses ; — the 
ratio  of  which  he  also  ascertained.  From  the  phenomena  obser 
ved  in  these  experiments,  Newton  deduced  his  Theory  of  Fits  of 
EASY  REFLECTION  AND  TRANSMISSION  of  light.  It  consists  in  suppos 
ing  that  every  particle  of  light,  from  its  first  discharge  from  a  lumi 
nous  body,  possesses,  at  equally  distant  intervals,  dispositions  to 
be  reflected  from,  or  transmitted  through  the  surfaces  of  bodies 
upon  which  it  may  fall.  For  instance,  if  the  rays  are  in  a  Fit  of 
Easy  Reflection,  they  are  on  reaching  the  surface,  repelled, 
thrown  off]  or  reflected  from  it ;  if,  in  a  Fit  of  Easy  Transmission, 
they  are  attracted,  drawn  in,  or  transmitted  through  it.  By  this 
Theory  of  Fits,  our  author  likewise  explained  the  colours  of 
thick  plates. 

He  regarded  light  as  consisting  of  small  material  particles 
emitted  from  shining  substances.  He  thought  that  these  parti 
cles  could  be  re-combined  into  solid  matter,  so  that  "  gross  bodies 
and  light,  were  convertible  into  one  another  ;"  that  the  particles  of 
light  and  the  particles  of  solid  bodies  acted  mutually  upon  each 
other ;  those  of  light  agitating  and  heating  those  of  solid  bodies, 
and  the  latter  attracting  and  repelling  the  former.  Newton  was 
the  first  to  suggest  the  idea  of  the  POLARIZATION  of  light. 

In  the  paper  entitled  An  Hypothesis  Explaining  Properties  of 
Light,  December,  1675,  our  author  first  introduced  his  opinions  re 
specting  Ether — opinions  which  he  afterward  abandoned  and  again 


26  LIFE    OF    SIR      S.\AC    1SEWTON. 

permanently  resumed — "  A  most  subtle  spirit  which  pervades"  ah 
bodies,  and  is  expanded  through  all  the  heavens.  It  is  electric, 
and  almost,  if  not  quite  immeasurably  elastic  and  rare.  "  By  the 
force  and  action  of  which  spirit  the  particles  of  bodies  mutually 
attract  one  another,  at  near  distances,  and  cohere,  if  contiguous  ; 
and  electric  bodies  operate  at  greater  distances,  as  well  repelling 
as  attracting  the  neighbouring  corpuscles  ;  and  light  is  emitted, 
-reflected,  refracted,  inflected  and  heats  bodies  ;  and  all  sensation 
is  excited,  and  the  members  of  animal  bodies  move  at  the  com 
mand  of  the  will,  namely,  by  the  vibrations  of  this  spirit,  mutu 
ally  propagated  along  the  solid  filaments  of  the  nerves,  from  the 
outward  organs  of  sense  to  the  brain,  and  from  the  brain  into  the 
muscles."  This  "  spirit"  was  no  anima  mundi  ;  nothing  further 
from  the  thought  of  Newton  ;  but  was  it  not,  on  his  part,  a  par 
tial  recognition  of,  or  attempt  to  reach  an  ultimate  material  force, 
or  primary  element,  by  means  of  which,  "  in  the  roaring  loom  of 
time,"  this  material  universe,  God's  visible  garment,  may  be 
woven  for  us  ? 

The  Royal  Society  were  greatly  interested  in  the  results  of 
some  experiments,  which  our  author  had,  at  the  same  time,  com 
municated  to  them  relative  to  the  excitation  of  electricity  in  glass  ; 
and  they,  after  several  attempts  and  further  direction  from  him, 
succeeded  in  re-producing  the  same  phenomena. 

One  of  the  most  curious  of  Newton's  minor  inquiries  related  to 
the  connexion  between  the  refractive  powers  and  chemical  com 
position  of  bodies.  He  found  on  comparing  the  refractive  powers 
and  the  densities  of  many  different  substances,  that  the  former 
were  very  nearly  proportional  to  the  latter,  in  the  same  bodies. 
Unctuous  and  sulphureous  bodies  were  noticed  as  remarkable  excep 
tions — as  well  as  the  diamond — their  refractive  powers  being  two 
or  three  times  greater  in  respect  of  their  densities  than  in  the 
case  of  other  substances,  while,  as  among  themselves,  the  one  was 
generally  proportional  to  the  other.  He  hence  inferred  as  to  the 
diamond  a  great  degree  of  combustibility  •; — a  conjecture  which 
the  experiments  of  modern  chemistry  have  shown  to  be  true. 

The  chemical  researches  of  our  author  were  probably  pursued 
with  more  or  less  diligence  from  the  time  of  his  witnessing  some 


LIFE    OF    .SIR    ISAAC    NEWTON.  27 

?t'  the  uractical  operations  in  that  science  at  the  Apothecary's  at 
Grantham.     DE  NATURA  ACIDORUM  is  a  short  chemical  paper,  on 
various   topics,   and  published   in  Dr.   Horsley's  Edition  of  his 
works.     TABULA  QUANTITATUM  E  r  GRADUUM  COLORIS  was  in 
serted  iii  the  Philosophical  Transactions  ;  it  contains  a  compara 
tive  scale  of  temperature  from  that  of  melting  ice  to  that  of  a 
small  kitchen  coal-fire.     He  regarded  fire  as  a  body  heated  so  hot 
as  to  emit  light  copiously ;  and  flame  as  a  vapour,  fume,  or  ex 
halation  heated   so  hot  as  to  shine.     To  elective  attraction,  by 
the  operation  of  which  the  small  particles  of  bodies,  as  he  con 
ceived,  act  upon  one  another,  at  distances  so  minute  as  to  escape 
observation,  he  ascribed  all  the   various   chemical  phenomena  ot 
precipitation,  combination,  solution,  and   crystallization,   and   the 
mechanical  phenomena  of  cohesion  and  capillary  attraction.    New 
ton's  chemical  views  were   illustrated  and  confirmed,  in  part,  at 
least,  in  his  own  life-time.     As  to  the  structure  of  bodies,  he  was 
of  opinion  "  that  the  smallest  particles  of  matter   may  cohere  by 
the  strongest  attractions,  and  compose  bigger  particles  of  weaker 
virtue  ;  and  many  of  these  may  cohere  and  compose  bigger  par 
tides  whose  virtue  is  still  weaker ;  and  so  on  for  divers  succes 
sions,  until  the  progression  end  in  the  biggest  particles,  on  which 
the  operations  in  chemistry  and  the  colours  of  natural  bodies  de 
pend,  and  which  by  adhering,  compose  bodies  of  sensible  magni 
tude." 

There  is  good  reason  to  suppose  that  our  author  was  a  diligent 
student  of  the  writings  of  Jacob  Behmen  ;  and  that  in  conjunction 
with  a  relative,  Dr.  Newton,  he  was  busily  engaged,  for  several 
months  in  the  earlier  part  of  life,  in  quest  of  the  philosopher's 
tincture.  "  Great  Alchymist,"  however,  very  imperfectly  de 
scribes  the  character  of  Behmen,  whose  researches  into  things 
material  and  things  spiritual,  things  human  and  things  divine,  ai- 
ford  the  strongest  evidence  of  a  great  and  original  mind. 

More  appropriately  here,  perhaps,  than  elsewhere,  may  be 
given  Newton's  account  of  some  curious  experiments,  made  in  his 
own  person,  on  the  action  of  light  upon  the  retina,  Locke,  who 
was  an  intimate  friend  of  our  author,  wrote  to  him  for  his  opinion 
on  a  certain  fact  stated  in  Boyle's  Book  of  Colours.  Newton,  in 


2S  LIFE    OF    SIR    ISAAC    NEWTON. 

his  reply,  dated  June  30th,  16(Jl,  narrates  the  following  circum 
stances,  which  probably  took  place  in  the  course  of  his  optical 
researches.  Thus : — 

"  The  observation  you  mention  in  Mr.  Boyle's  Book  of  Colours 
I  once  tried  upon  myself  with  the  hazard  of  my  eyes.  The 
manner  was  this ;  I  looked  a  very  little  while  upon  the  sun  in  the 
looking-glass  with  my  right  eye,  and  then  turned  my  eyes  into  a 
dark  corner  of  my  chamber,  arid  winked,  to  observe  the  impres 
sion  made,  and  the  circles  of  colours  which  encompassed  it,  and 
how  they  decayed  by  degrees,  and  at  last  vanished.  This  I  re 
peated  a  second  and  a  third  time.  At  the  third  time,  when  the 
phantasm  of  light  and  colours  about  it  were  almost  vanished,  in 
tending  my  fancy  upon  them  to  see  their  last  appearance,  I  found, 
to  my  amazement,  that  they  began  to  return,  and  by  little  and 
little  to  become  as  lively  and  vivid  as  when  I  had  newly  looked 
upon  the  sun.  But  when  I  ceased  to  intend  my  fancy  upon  them, 
they  vanished  again.  After  this,  I  found,  that  as  often  as  I  went 
into  the  dark,  and  intended  my  mind  upon  them,  as  when  a  man 
looks  earnestly  to  see  anything  which  is  difficult  to  be  seen,  I 
could  make  the  phantasm  return  without  looking  any  more  upon 
the  sun ;  and  the  oftener  I  made  it  return,  the  more  easily  I  could 
make  it  return  again.  And,  at  length,  by  repeating  this,  without 
looking  any  more  upon  the  sun,  I  made  such  an  impression  on  my 
eye,  that,  if  I  looked  upon  the  clouds,  or  a  book,  or  any  bright 
object,  I  saw  upon  it  a  round  bright  spot  of  light  like  the  sun, 
and,  which  is  still  stranger,  though  I  looked  upon  the  sun  with 
my  right  eye  only,  and  not  with  my  left,  yet  my  fancy  began  *o 
make  an  impression  upon  my  left  eye,  as  well  us  upon  my  right. 
For  if  I  shut  my  right  eye,  or  looked  upon  a  book,  or  the  clouds, 
with  my  left  eye,  I  could  see  the  spectrum  of  the  sun  almost  as 
plain  as  with  my  right  eye,  if  I  did  but  intend  my  fancy  a  little 
while  upon  it ;  for  at  first,  if  I  shut  my  right  eye,  and  looked  with 
my  left,  the  spectrum  of  the  sun  did  not  appear  till  I  intended  my 
fancy  upon  it ;  but  by  repeating,  this  appeared  every  time  more 
easily.  And  now,  in  a  few  hours'  time,  I  had  brought  my  eyes 
to  such  a  pass,  that  I  could  look  upon  no  blight  object  with  either 
eye,  but  I  saw  the  sun  before  me,  so  that  I  durst  neither  write 


LIFE    OF    SIR    ISAAC    NEWTON.  29 

nor  read  ;  but  to  recover  the  use  of  my  eyes,  shut  myself  up  in 
my  chamber  made  dark,  for  three  days  together,  and  used  all 
means  to  divert  my  imagination  from  the  sun.  For  if  I  thought 
upon  him,  I  presently  saw  his  picture,  though  I  was  in  the  dark. 
But  by  keeping  in  the  dark,  and  employing  my  mind  about  other 
things,  I  began  in  three  or  four  days  to  have  some  use  of  my  eyes 
again ;  and  by  forbearing  to  look  upon  bright  objects,  recovered 
them  pretty  well,  though  not  so  well  but  that,  for  some  months 
after,  the  spectrum  of  the  sun  began  to  return  as  often  as  I  began 
to  meditate  upon  the  phenomena,  even  though  I  lay  in  bed  at  mid 
night  with  my  curtains  drawn.  But  now  I  have  been  very  well 
for  many  years,  though  I  am  apt  to  think,  if  I  durst  venture  my 
eyes,  I  could  still  make  the  phantasm  return  by  the  power  of  my 
fancy.  This  story  I  tell  you,  to  let  you  understand,  thaj;  in  the 
observation  related  by  Mr.  Boyle,  the  man's  fancy  probably  con 
curred  with  the  impression  made  by  the  sun's  light  to  produce 
that  phantasm  of  the  sun  which  he  constantly  saw  in  bright  ob 
jects.  And  so  your  question  about  the  cause  of  phantasm  in 
volves  another  about  the  power  of  fancy,  which  I  must  confess  is 
too  hard  a  knot  for  me  to  untie.  To  place  this  effect  in  a  constant 
motion  is  hard,  because  the  sun  ought  then  to  appear  perpetually. 
It  seems  rather  to  consist  in  a  disposition  of  the  sensorium  to 
move  the  imagination  strongly,  and  to  be  easily  moved,  both  by 
the  imagination  and  by  the  light,  as  often  as  bright  objects  are 
looked  upon."J 

Though  Newton  had  continued  silent,  yet  his  thoughts  were 
by  no  means  inactive  upon  the  vast  subject  of  the  planetary  mo 
tions.  The  idea  of  Universal  Gravitation,  first  caught  sight  of,  so 
to  speak,  in  the  garden  at  Woolsthorpe,  years  ago,  had  gradually 
expanded  upon  him.  We  find  him,  in  a  letter  to  Dr.  Hooke, 
Secretary  of  the  Royal  Society,  dated  in  November,  1679,  pro 
posing  to  verify  the  motion  of  the  earth  by  direct  experiment, 
namely,  by  the  observation  of  the  path  pursued  by  a  body  falling 
from  a  considerable  height.  He  had  concluded  that  the  path 
would  be  spiral ;  but  Dr.  Hooke  maintained  that  it  would  be  an 
eccentric  ellipse  iu  vacuo,  and  an  ellipti-spiral  in  a  resisting  me 
dium.  Our  author,  aided  by  this  correction  of  his  error,  and  by 


30  LIFE    OF    SIR    ISAAC    NEWTON. 

the  discovery  that  a  projectile  would  move  in  an  elliptical  orbil 
when  under  the  influence  of  a  force  varying  inversely  as  the 
square  of  the  distance,  was  led  to  discover  "  the  theorem  bj 
which  he  afterwards  examined  the  ellipsis ;"  and  to  demonstrate 
the  celebrated  proposition  that  a  planet  acted  upon  by  an  attrac 
tive  force  varying  inversely  as  the  squares  of  the  distances  will 
describe  an  elliptical  orbit,  in  one  of  whose  foci  the  attractive 
force  resides. 

When  he  was  attending  a  meeting  of  the  Royal  Society,  in 
June  1682,  the  conversation  fell  upon  the  subject  of  the  measure 
ment  of  a  degree  of  the  meridian,  executed  by  M.  Picard,  a 
French  Astronomer,  in  1679.  Newton  took  a  memorandum  oi 
the  result ;  and  afterward,  at  the  earliest  opportunity,  computed 
from  it  the  diameter  of  the  earth  :  furnished  with  these  new  data, 
he  resumed  his  calculation  of  1666.  As  he  proceeded  therein, 
he  saw  that  his  early  expectations  were  now  likely  to  be  realized  ; 
the  thick  rushing,  stupendous  results  overpowered  him ;  he  be 
came  unable  to  carry  on  the  process  of  calculation,  and  intrusted 
its  completion  to  one  of  his  friends.  The  discoverer  had,  indeed, 
grasped  the  master-fact.  The  law  of  falling  bodies  at  the  earth's 
surface  was  at  length  identified  with  that  which  guided  the  moon 
in  her  orbit.  And  so  his  GREAT  THOUGHT,  that  had  for  sixteen 
years  loomed  up  in  dim,  gigantic  outline,  amid  the  first  dawn  of  a 
plausible  hypothesis,  now  stood  forth,  radiant  and  not  less  grand, 
in  the  mid-day  light  of  demonstrated  truth. 

It  were  difficult,  nay  impossible  to  imagine,  even,  the  influence 
of  a  result  like  this  upon  a  mind  like  Newton's.  It  was  as  if  the 
keystone  had  been  fitted  to  the  glorious  arch  by  which  his  spirit 
should  ascend  to  the  outskirts  of  infinite  space — spanning  the  immea 
surable — weighing  the  imponderable — computing  the  incalculable 
—mapping  out  the  marchings  of  the  planets,  and  the  far-wander 
ings  of  the  comefs,  and  catching,  bring  back  to  earth  some  clearer 
notes  of  that  higher  melody  which,  as  a  sounding  voice,  bears 
perpetual  witness  to  the  design  and  omnipotence  of  a  creating 
Deity. 

Newton,  extending  the  law  thus  obtained,  composed  a  series 
of  about  twelve  propositions  on  the  motion  of  the  primary  planets 


LIFE    OF    SIR    ISAAC    NEWTON.  31 

about  the  sun.  These  were  sent  to  London,  and  communicated 
to  the  Royal  Society  about  the  end  of  1683.  At  or  near  this  pe 
riod,  other  philosophers,  as  Sir  Christopher  Wren,  Dr.  Halley, 
and  Dr.  Hooke,  were  engaged  in  investigating  the  same  subject ; 
but  with  no  definite  or  satisfactory  results.  Dr.  Halley,  having 
seen,  it  is  presumed,  our  author's  propositions,  went  in  August, 
1684,  to  Cambridge  to  consult  with  him  upon  the  subject. 
Newton  assured  him  that  he  had  brought  the  demonstration  to 
perfection.  In  November,  Dr.  Halley  received  a  copy  of  the 
work  ;  and,  in  the  following  month^  announced  .  it  to  the  Royal 
Society,  with  the  author's  promise  to  have  it  entered  upon  their 
Register.  Newton,  subsequently  reminded  by  the  Society  of  his 
promise,  proceeded  in  the  diligent  preparation  of  the  work,  and. 
though  suffering  an  interruption  of  six  weeks,  transmitted  the 
manuscript  of  the  first  book  to  London  before  the  end  of  April. 
The  work  was  entitled  PHILOSOPHI/E  NATURALIS  PRINCIPIA 
MATHEMATICA,  dedicated  to  the  Royal  Society,  and  presented 
thereto  on  the  28th  of  April,  1685-6.  The  highest  encomiums 
were  passed  upon  it ;  and  the  council  resolved,  on  the  19th  of 
May,  to  print  it  at  the  expense  of  the  Society,  and  under  the  di 
rection  of  Dr.  Halley.  The  latter,  a  few  days  afterward,  com 
municated  these  steps  to  Newton,  who,  in  a  reply,  dated  the  20th 
of  June,  holds  the  following  language  : — "  The  proof  you  sent  me 
I  like  very  well.  I  designed  the  whole  to  consist  of  three  books  ; 
the  second  was  finished  last  summer,  being  short,  and  only  wants 
transcribing,  and  drawing  the  cuts  fairly.  Some  new  propositions 
I  have  since  thought  on,  which  I  can  as  well  let  alone.  The 
third  wants  the  theory  of  comets.  In  autumn  last,  I  spent  two 
months  in  calculation  to  no  purpose  for  want  of  a  good  method, 
which  made  me  afterward  return  to  the  first  book,  and  enlarge  it 
with  diverse  propositions,  some*  relating  to  comets,  others  to  other 
things  found  ouf  last  winter.  The  third  I  now  design  to  sup 
press.  Philosophy  is  such  an  impertinently  litigious  lady,  that  a 
man  had  as  good  be  engaged  in  l«iw-suits  as  have  to  do  with  her. 
I  found  it  so  formerly,  and  now  I  can  no  sooner  come  near  her 
again,  but  she  gives  me  warning.  The  first  two  books  without 
the  third  will  not  so  well  bear  the  title  of  P/iilosophicc  Naturalis 

3 


32  LIFE    OF    SIR    ISAAC    NEWTON. 

Principia  Mathematicia  ;  and  thereupon  I  had  altered  it  to  this, 
De  Motu  Corporum  Libri  duo.  But  after  second  thought  I  re 
tain  the  former  title.  It  will  help  the  sale  of  the  book,  which  I 
ought  not  to  diminish  now  'tis  yours." 

This  "  warning"  arose  from  some  pretensions  put  forth  by  Dr. 
Hooke.  And  though  Newton  gave  a  minute  and  positive  refuta 
tions  of  such  claims,  yet,  to  reconcile  all  differences,  he  gener 
ously  added  to  Prop.  IV.  Cor.  6,  Book  I,  a  Scholium,  in  which 
Wren,  Hooke  and  Halley  are  acknowledged  to  have  indepen 
dently  deduced  the  law  of  gravity  from  the  second  law  of 
Kepler. 

The  suppression  of  the  third  book  Dr.  Halley  could  not  endure 
to  see.  "  I  must  again  beg  you"  says  he,  "  not  to  let  your  re 
sentments  run  so  high  as  to  deprive  us  of  your  third  book,  where 
in  your  applications  of  your  mathematical  doctrine  to  the  theory 
of  comets,  and  several  curious  experiments,  which,  as  I  guess  by 
what  you  write  ought  to  compose  it,  will  undoubtedly  render  it 
acceptable  to  those  who  will  call  themselves  philosophers  without 
mathematics,  which  are  much  the  greater  number."  To  these 
solicitations  Newton  yielded.  There  were  no  "resentments,"  how 
ever,  as  we  conceive,  in  his  "  design  to  suppress."  He  sought 
peace  ;  for  he  loved  and  valued  it  above  all  applause.  But,  in 
spite  of  his  efforts  for  tranquillity's  sake,  his  course  of  discovery 
was  all  along  molested  by  ignorance  or  presumptuous  rivalry. 

The  publication  of  the  great  work  now  went  rapidly  forwards, 
The  second  book  was  sent  to  the  Society,  and  presented  on  the 
2d  March  ;  the  third,  on  the  6th  April ;  and  the  whole  was  com 
pleted  and  published  in  the  month  of  May,  1686-7.  In  the  sec 
ond  Lemma  of  the  second  book,  the  fundamental  principle  of  his 
fiuxionary  calculus  was,  for  the  first  time,  given  to  the  world ;  but 
its  algorithm  or  notation  did  not  appear  till  published  in  the 
second  volume  nf  Dr.  Wallis's  works,  in  1693. 

And  thus  was  ushered  into  existence  The  PRINCIPIA — a  work 
to  which  pre-eminence  above  all  the  productions  of  the  human 
intellect  has  been  awarded — a  work  that  must  be  esteemed  of 
priceless  worth  so  long  as  Science  has  a  votary,  or  a  single  wor 
shipper  be  left  to  kneel  at  the  altar  of  Truth. 


LIFE    OF    SIR    ISAAC    NEWTON.  33 

The  entire  work  bears  the  general  title  of  THE  MATHEMATICAL 
PRINCIPLES  OF  NATURAL  PHILOSOPHY.  It  consists  of  three  books: 
the  first  two,  entitled,  OF  THE  MOTION  OF  BODIES,  are  occupied 
with  the  laws  and  conditions  of  motions  and  forces,  and  are  illus 
trated  with  many  scholia  treating  of  some  of  the  most  general 
and  best  established  points  in  philosophy,  such  as  the  density  and 
resistance  of  bodies,  spaces  void  of  matter,  and  the  motion  of 
sound  and  light.  From  these  principles,  there  is  deduced,  in  the 
third  book,  drawn  up  in  as  popular  a  style  as  possible  and  entitled, 
OF  THE  SYSTEM  OF  THE  WORLD,  the  constitution  of  the  system  of 
i he  world.  In  regard  to  this  book,  the  author  say^  — "  I  had,  indeed, 
composed  the  third  Book  in  a  popular  method,  that  it  might  be  read 
by  many ;  but  afterwards,  considering  that  such  as  had  not  suf- 
ficently  entered  into  the  principles  could  not  easily  discover  the 
strength  of  the  consequences,  nor  lay  aside  the  prejudices  to  which 
they  had  been  many  years  accustomed,  therefore,  to  prevent  dis 
putes  which  might  be  raised  upon  such  accounts,  I  chose  to  reduce 
the  substance  of  this  Book  into  the  form  of  Propositions  (in  the 
mathematical  way),  which  should  be  read  by  those  only  who  had 
first  made  themselves  masters  of  the  principles  established  in  the 
preceding  Books  :  not  that  I  would  advise  any  one  to  the  previous 
study  of  every  Proposition  of  those  Books." — "It  is  enough  it 
one  carefully  reads  the  Definitions,  the  Laws  of  Motion,  and  the 
three  first  Sections  of  the  first  Book.  He  may  then  pass  on  to 
this  Book,  and  consult  such  of  the  remaining  Propositions  of  the 
first  two  Books,  as  the  references  in  this,  and  his  occasions  shall  re 
quire."  So  that  "  The  System  of  the  World"  is  composed  both 
"  in  a  popular  method,"  and  in  the  form  of  mathematical  Propo 
sitions. 

The  principle  of  Universal  Gravi'  ition,  namely,  that  every 
particle  of  matter  is  attracted  by,  or  gravitates  to,  every  other 
particle  of  matter,  icith  a  force  inversely  proportional  to  the 
squares  of  their  distances — is  the  discovery  w?  ich  characterizes 
The  PRINCIPIA.  This  principle  the  author  deduced  from  the  mo 
tion  of  the  moon,  and  the  three  laws  of  Kepler — laws,  which 
Newton,  in  turn,  by  his  greater  law,  demonstrated  to  be  true. 

From  the  first  law  of  Kepler,  namely,  the   proportionality  of 


LIFE    OF    SIR    ISAAC    NEWTON. 

the  areas  to  t\ie  times  of  their  description,  our  author  inferred 
that  the  force  which  retained  the  planet  in  its  orbit  was  always 
directed  to  the  sun ;  and  from  the  second,  namely,  that  every 
planet  moves  in  an  ellipse  with  the  sun  in  one  of  its  foci,  he  drew 
the  more  general  inference  that  the  force  by  which  the  planet 
moves  round  that  focus  varies  inversely  as  the  square  of  its  dis 
tance  therefrom  :  and  he  demonstrated  that  a  planet  acted  upon 
by  such  a  force  could  not  move  in  any  other  curve  than  a  conic 
section ;  showing  when  the  moving  body  would  describe  a  circu 
lar,  an  elliptical,  a  parabolic,  or  hyperbolic  orbit.  He  demon 
strated,  too,  that  this  force,  or  attracting,  gravitating  power  re 
sided  in  every,  the  least  particle ;  but  that,  in  spherical  masses,  it 
operated  as  if  confined  to  their  centres ;  so  that,  one  sphere  or 
body  will  act  upon  another  sphere  or  body,  with  a  force  directly 
proportional  to  the  quantity  of  matter,  and  inversely  as  the  square 
of  the  distance  between  their  centres;  and  that  their  velocities  of 
mutual  approach  will  be  in  the  inverse  ratio  of  their  quantities  o*' 
matter.  Thus  he  grandly  outlined  the  Universal  Law.  Verify 
ing  its  truth  by  the  motions  of  terrestrial  bodies,  then  by  those  of 
the  moon  and  other  secondary  orbs,  he  finally  embraced,  in  one 
mighty  generalization,  the  entire  Solar  System — all  the  move 
ments  of  all  its  bodies — planets,  satellites  and  comets — explain 
ing  and  harmonizing  the  many  diverse  and  theretofore  inexplica 
ble  phenomena. 

Guided  by  the  genius  of  Newton,  we  see  sphere  bound  to 
sphere,  body  to  body,  particle  to  particle,  atom  to  mass,  the  min 
utest  part  to  the  stupendous  whole — each  to  each,  each  to  all, 
and  all  to  each — in  the  mysterious  bonds  of  a  ceaseless,  recipro 
cal  influence.  An  influence  whose  workings  are  shown  to  be 
alike  present  in  the  globular  dew-drop,  or  oblate-spheroidal  earth  ; 
in  the  falling  shower,  or  vast  heaving  ocean  tides ;  in  the  flying 
thistle-down,  or  fixed,  ponderous  rock  ;  in  the  swinging  pendulum, 
or  time-measuring  sun  ;  in  the  varying  and  unequal  moon,  or 
earth's  slowly  retrograding  poles  ;  in  the  uncertain  meteor,  or 
oiazing  comet  wheeling  swiftly  away  on  its  remote,  yet  determined 
round.  An  influence,  in  fine,  that  may  link  system  to  system 
through  all  the  star-glowing  firmament ;  then  firmament  to  iirma- 


LIFE    OF    SIR    ISAAC    NEWTON.  35 

merit ;  aye,  firmament  to  firmament,  again  and  again,  till,  con 
verging  home,  it  may  be,  to  some  ineffable  centre,  where  more 
presently  dwells  He  who  inhabiteth  immensity,  and  where  infini 
tudes  meet  and  eternities  have  their  condux,  and  where  around 
move,  in  softest,  swiftest  measure,  all  the  countless  hosts  that 
crowd  heaven's  fathomless  deeps. 

And  yet  Newton,  amid  the  loveliness  and  magnitude  of  Om 
nipotence,  lost  not  sight  of  the  Almighty  One.  A  secondary, 
however  universal,  was  not  taken  for  the  First  Cause.  An  im 
pressed  force,  however  diffused  and  powerful,  assumed  not  the 
functions  of  the  creating,  giving  Energy.  Material  beauties, 
splendours,  and  sublimities,  however  rich  in  glory,  and  endless  in 
extent,  concealed  not  the  attributes  of  an  intelligent  Supreme. 
From  the  depths  of  his  own  soul,  through  reason  and  the  WORD, 
he  had  risen,  a  priori,  to  God  :  from  the  heights  of  Omnipotence, 
through  the  design  and  law  of  the  builded  universe,  he  proved  </ 
posteriori,  a  Deity.  "  I  had,"  says  he,  "  an  eye  upon  such  prin 
ciples  as  might  work,  with  considering  men,  for  the  belief  of  a 
Deity,"  in  writing  the  PRINCIPIA  ;  at  the  conclusion  whereof,  he 
teaches  that — "  this  most  beautiful  system  of  the  sun,  planets  and 
comets,  could  only  proceed  from  the  counsel  and  dominion  of  an 
intelligent  and  powerful  Being.  And  if  the  fixed  stars  are  the 
centres  of  other  like  systems,  these,  being  forme  1  by  the  like 
wise  counsels,  must  be  all  subject  to  the  dominion  of  One ;  especially 
since  the  light  of  the  fixed  stars  is  of  the  same  nature  with  the 
light  of  the  sun,  and  from  every  system  light  passes  into  all  other 
systems  :  and  lest  the  systems  of  the  fixed  stars  should,  by  their 
gravity,  fall  on  each  other  mutually,  he  hath  placed  those  systems 
at  immense  distances  one  from  another. 

"  This  Being  governs  all  things,  not  as  the  soul  of  the  world, 
but  as  Lord  over  all ;  and  on  account  of  his  dominion  he  is  wont, 
to  be  called  Lord  God  Travrowparwp  or  Universal  Ruler ;  for  God 
is  a  relative  word,  and  has  a  respect  to  servants  ;  and  Deity  is 
the  dominion  of  God,  not  over  his  own  body,  as  those  imagine 
who  fancy  God  to  be  the  soul  of  the  world,  but  over  servants. 
The  Supreme  God  is  a  Being  eternal,  infinite,  absolutely  perfect ; 
but  a  being,  however  perfect,  without  dominion,  cannot  be  said  to 


36  LIFE    OF    SIR    ISAAC    NEWTON. 

be  Lord  God ;  for  we  say,  my  God,  your  God,  the  God  of  Israel 
the  God  of  Gods,  and  Lord  of  Lords  ;  but  we  do  not  say,  my 
Eternal,  your  Eternal,  the  Eternal  of  Israel,  the  Eternal  of  Gods  : 
we  do  not  say  my  Infinite,  or  my  Perfect :  these  are  titles  which 
have  no  respect  to  servants.  The  word  God  usually  signifies 
Lord  ;  but  every  Lord  is  not  God.  It  is  the  dominion  of  a  spir 
itual  Being  which  constitutes  a  God  ;  a  true,  supreme,  or  imagi 
nary  dominion  makes  a  true,  supreme,  or  imaginary  God.  And 
from  his  true  dominion  it  follows  that  the  true  God  is  a  living, 
intelligent  and  powerful  Being ;  and  from  his  other  perfections, 
that  he  is  supreme  or  most  perfect.  He  is  eternal  and  in 
finite,  omnipotent  and  omniscient ;  that  is,  his  duration  reaches 
from  eternity  to  eternity  ;  his  presence  from  infinity  to  infinity ; 
he  governs  all  things  and  knows  all  things,  that  are  or  can  be 
done.  He  is  not  eternity  or  infinity,  but  eternal  and  infinite  ; 
he  is  not  duration  and  space,  but  he  endures  and  is  present. 
He  endures  forever  and  is  everywhere  present ;  and  by  existing 
always  and  everywhere,  he  constitutes  duration  and  space.  Since 
every  particle  of  space  is  always,  and  every  indivisible  moment 
of  duration  is  everywhere,  certainly  the  Maker  and  Lord  of  things 
cannot  be  never  and  nowhere.  Every  soul  that  has  perception 
is,  though  in  different  times  and  different  organs  of  sense  and  mo 
tion,  still  the  same  indivisible  person.  There  are  given  succes 
sive  parts  in  duration,  co-existent  parts  in  space,  but  neither  the 
one  nor  the  other  in  the  person  of  a  man,  or  his  thinking 
principle ;  and  much  less  can  they  be  found  in  the  thinking  sub 
stance  of  God.  Every  man.  so  far  as  he  is  a  thing  that  has  j:er- 
ceptiori,  is  one  and  the  same  man  during  his  whole  life,  in  all  and 
each  of  his  organs  of  sense.  God  is  one  and  the  same  God,  al 
ways  and  everywhere.  He  is  omnipresent,  not  virtually  only, 
but  also  substantially ;  for  virtue  cannot  subsist  without  sub 
stance.  In  him  are  all  things  contained  and  moved ;  yet  neither 
affects  the  other  ;  God  suffers  nothing  from  the  motion  of  bodies  ; 
bodies  find  no  resistance  from  the  omnipresence  of  God.  It  is 
allowed  by  all  that  the  Supreme  God  exists  necessarily  ;  and  by 
the  same  necessity  he  exists  always  and  everywhere.  Whence 
also  he  is  all  similar,  all  eye,  all  ear,  all  brain,  all  arm,  all  powei 


LIFE    CF    SIR    ISAAC    NEWTON.  37 

to  perceive,  to  understand,  and  to  act ;  but  in  a  manner  not  at  all 
human,  in  a  manner  not  at  all  corporeal,  in  a  manner  utterly  un 
known  to  us.  As  a  blind  man  has  no  idea  of  colours,  so  have  we 
no  idea  of  the  manner  by  which  the  all-wise  God  perceives  and 
understands  all  things.  He  is  utterly  void  of  all  body,  and  bodily 
figure,  and  can  therefore  neither  be  seen,  nor  heard,  nor  touched  ; 
nor  ought  he  to  be  worshipped  under  the  representation  of  any 
corporeal  thing.  We  have  ideas  of  his  attributes,  but  what  the 
real  substance  of  anything  is  we  know  not.  In  bodies  we  see 
only  their  figures  and  colours,  we  hear  only  the  sounds,  we  touch 
only  their  outward  surfaces,  we  smell  only  the  smells,  and  taste 
only  the  savours  ;  but  their  inward  substances  are  not  to  be  known, 
either  by  our  senses,  or  by  any  reflex  act  of  our  minds  :  much 
less,  then,  have  we  any  idea  of  the  substance  of  God.  We  know 
him  only  by  his  most  wise  and  excellent  contrivances  of  things, 
and  final  causes  ;  we  admire  him  for  his  perfections  ;  but  we  rev 
erence  and  adore  him  on  account  of  his  dominion ;  for  we  adore 
him  as  his  servants  ;  and  a  god  without  dominion,  providence,  and 
final  causes,  is  nothing  else  but  Fate  and  Nature.  Blind  meta 
physical  necessity,  which  is  certainly  the  same  always  and  every 
where,  could  produce  no  variety  of  things.  All  that  diversity  of 
natural  things  which  we  find  suited  to  different  times  and  places 
could  arise  from  nothing  but  the  ideas  and  will  of  a  Being  neces 
sarily  existing." 

Thus,  the  diligent  student  of  science,  the  earnest  seeker  of 
truth,  led,  as  through  the  courts  of  a  sacred  Temple,  wherein,  at 
each  step,  new  wonders  meet  the  eye,  till,  as  a  crowning  grace, 
they  stand  before  a  Holy  of  Holies,  and  learn  that  all  science  and 
all  truth  are  one  which  hath  its  beginning  and  its  end  in  the 
knowledge  of  Him  whose  glory  the  heavens  declare,  and  whose 
handiwork  the  firmament  showeth  forth. 

The  introduction  of  the  pure  and  lofty  doctrines  of  the  PRIN- 
CIPIA  was  perseveringly  resisted.  Descartes, with  his  system  of 
vortices,  had  sown  plausibly  to  the  imagination,  and  error  had 
struck  down  deeply,  and  shot  up  luxuriantly,  not  only  in  the 
popular,  but  in  the  scientific  mind.  Besides  the  idea — in  itself  so 
simple  and  so  grand — that  the  great  masses  of  the  planets  were 


38  LIFE    OF    SIR    ISAAC    NEWTON. 

suspended  in  empty  space,  and  retained  in  their  orbits  by  an  in 
visible  influence  residing  in  the  sun — was  to  the  ignorant  a  thing 
inconceivable,  and  to  the  learned  a  revival  of  the  occult  qualities 
of  the  ancient  physics.  This  remark  applies  particularly  to  the 
continent.  Leibnitz  misapprehended  ;  Huygens  in  part  rejected  ; 
John  Bernouilli  opposed  ;  and  Fontenelle  never  received  the  doc 
trines  of  the  PRINCIPIA.  So  that,  the  saying  of  Voltaire  is  prob 
ably  true,  that  though  Newton  survived  the  publication  of  his 
great  work  more  than  forty  years,  yet,  at  the  time  of  his  death, 
lie  had  not  above  twenty  followers  out  of  England. 

But  in  England,  the  reception  of  our  author's  philosophy  was 
rapid  and  triumphant.  His  own  labours,  while  Lucasian  Pro 
fessor  ;  those  of  his  successors  in  that  Chair — Whiston  and 
Saunderson  ;  those  of  Dr.  Samuel  Clarke,  Dr.  Laughton,  Roger 
Cotes,  and  Dr.  Bentley  ;  the  experimental  lectures  of  Dr.  Keill 
and  Desaguliers  ;  the  early  and  powerful  exertions  of  David 
Gregory  at  Edinburgh,  and  of  his  brother  James  Gregory  at  St. 
Andrew's,  tended  to  diffuse  widely  in  England  and  Scotland  a 
knowledge  of,  and  taste  for  the  truths  of  the  PRINCIPIA.  Indeed, 
its  mathematical  doctrines  constituted,  from  the  first,  a  regular 
part  of  academical  instruction ;  while  its  physical  truths,  given  to 
the  public  in  popular  lectures,  illustrated  by  experiments,  had, 
before  the  lapse  of  twenty  )  (  ar.s,  become  familiar  to,  and  adopted 
by  the  general  mind.  Pemberton's  popular  "  View  of  Sir  Isaac 
Newton's  Philosophy"  was  published,  in  1728  ;  and  the  year  after 
ward,  an  English  translation  of  the  PRINCIPIA,  and  System  of  the 
World,  by  Andrew  Motte.  And  since  that  period,  the  labours  of 
Le  Seur  and  Jacquier,  of  Thorpe,  of  Jebb,  of  Wright  and  others 
have  greatly  contributed  to  display  the  most  hidden  treasures  of 
the  PRINCIPIA. 

About  the  time  of  the  publication  of  the  Principia,  James  II., 
bent  on  re-establishing  the  Romish  Faith,  had,  among  other  ille 
gal  acts,  ordered  by  mandamus,  the  University  of  Cambridge  to 
confer  the  degree  of  Master  of  Arts  upon  an  ignorant  monk. 
Obedience  to  this  mandate  was  resolutely  refused.  Newton  was 
one  of  the  nine  delegates  chosen  to  defend  the  independence  of 
the  University.  They  appeared  before  the  High  Court ; — and 


LIFE    OF    SIR    ISAAC    NEWTON.  39 

successfully :  the  king  abandoned  his  design.  The  prominent 
part  which  our  author  took  in  these  proceedings,  and  his  eminence 
in  the  scientific  world,  induced  his  proposal  as  one  of  the  parlia 
mentary  representatives  of  the  University.  He  was  elected,  in 
1688,  and  sat  in  the  Convention  Parliament  till  its  dissolution. 
After  the  first  year,  however,  he  seems  to  have  given  little  or  no 
attention  to  his  parliamentary  duties,  being  seldom  absent  from 
the  University  till  his  appointment  in  the  Mint,  in  1695. 

Newton  began  his  theological  researches  sometime  previous  to 
1691  ;  in  the  prime  of  his  years,  and  in  the  matured  vigour  of 
his  intellectual  powers.  From  his  youth,  as  we  have  seen,  he 
had  devoted  himself  with  an  activity  the  most  unceasing,  and  an 
energy  almost  superhuman  to  the  discovery  of  physical  truth  ; — 
giving  to  Philosophy  a  new  foundation,  and  to  Science  a  new 
temple.  To  pass  on,  then,  from  the  consideration  of  the  material, 
more  directly  to  that  of  the  spiritual,  was  a  natural,  nay,  with  so 
large  and  devout  a  soul,  a  necessary  advance.  The  Bible  was  to 
him  of  inestimable  worth.  In  the  elastic  freedom,  which  a  pure 
and  unswerving  faith  in  Him  of  Nazareth  gives,  his  mighty  facul 
ties  enjoyed  the  only  completest  scope  for  development.  His 
original  endowment,  however  great,  combined  with  a  studious 
application,  however  profound,  would  never,  without  this  libera 
tion  from  the  dominion  of  passion  and  sense,  have  enabled  him  to 
attain  to  that  wondrous  concentration  and  grasp  of  intellect,  for 
which  Fame  has  as  yet  assigned  him  no  equal.  Gratefully  he 
owned,  therefore,  the'same  Author  in  the  Book  of  Nature  and  the 
Book  of  Revelation.  These  were  to  him  as  drops  of  the  same 
unfathomable  ocean  ; — as  outrayings  of  the  same  inner  splendour  ; 
— as  tones  of  the  same  ineffable  voice  ; — as  segments  of  the  same 
infinite  curve.  "With  great  joy  he  had  found  himself  enabled  to 
proclaim,  as  an  interpreter,  from  the  hieroglyphs  of  Creation,  the 
existence  of  a  God  :  and  now,  with  greater  joy,  and  in  the  fulness 
of  his  knowledge,  and  in  the  fulness  of  his  strength,  he  laboured 
to  make  clear,  from  the  utterances  of  the  inspired  Word,  the  far 
mightier  confirmations  of  a  Supreme  Good,  in  all  its  glorious 
amplitude  of  Being  and  of  Attribute  ;  and  to  bring  the  infallible 
workings  thereof  plainly  home  to  the  understandings  and  the 


40  LIFE    OF    SIR    ISAAC    NEWTON. 

•affections  of  his  fellow-men ;  and  finally  to  add  the  weight  of  his 
own  testimony  in  favour  of  that  Religion,  whose  truth  is  now.  in 
deed,  "  girded  with  the  iron  and  the  rock  of  a  ponderous  and  co 
lossal  demonstration." 

His  work,  entitled,  OBSERVATIONS  UPON  THE  PROPHECIES  OF 
HOLY  WRIT,  PARTICULARLY  THE  PROPHECIES  OF  DANIEL  AND  THE 
APOCALYPSE  OF  ST.  JOHN,  first  published  in  London,  in  1733  4to. 
consists  of  two  parts  :  the  one  devoted  to  the  Prophecies  oi 
Daniel,  and  the  other  to  the  Apocalypse  of  St.  John.  In  the  first 
part,  he  treats  concerning  the  compilers  of  the  books  of  the  Old 
Testament ; — of  the  prophetic  language  ; — of  the  vision  of  the 
four  beasts ; — of  the  kingdoms  represented  by  the  feet  of  the 
image  composed  of  iron  and  clay  ; — of  the  ten  kingdoms  repre 
sented  by  the  ten  horns  of  the  beast ; — of  the  eleventh  horn  of 
Daniel's  fourth  beast ;  of  the  power  which  should  change  times 
and  laws  ;— of  the  kingdoms  represented  in  Daniel  by  the  ram 
and  he-goat ; — of  the  prophecy  of  the  seventy  weeks  ; — of  the 
times  of  the  birth  and  passion  of  Christ ; — of  the  prophecy  of  the 
Scripture  of  Truth  ; — of  the  king  who  doeth  according  to  his  will, 
and  magnified  himself  above  every  god,  and  honoured  Mahuzzims, 
and  regarded  not  the  desire  of  women ; — of  the  Mahuzzim,  hon 
oured  by  the  king  who  doeth  according  to  his  will.  In  the  sec 
ond  part,  he  treats  of  the  time  when  the  Apocalypse  was  written  , 
of  the  scene  of  the  vision,  and  the  relation  which  the  Apocalypse 
has  to  the  book  of  the  law  of  Moses,  and  to  the  worship  of  God 
in  the  temple  ; — of  the  relation  which  the  Apocalypse  has  to  the 
prophecies  of  Daniel,  and  of  the  subject  of  the  prophecy  itself 
Newton  regards  the  prophecies  as  given,  not  for  the  gratification 
of  man's  curiosity,  by  enabling  him  to  foreknow  ;  but  for  his  con 
viction  that  the  world  is  governed  by  Providence,  by  witnessing 
their  fulfilment.  Enough  of  prophecy,  he  thinks,  has  already 
been  fulfilled  to  afford  the  diligent  seeker  abundant  evidence  of 
God's  providence.  The  whole  work  is  marked  by  profound 
erudition,  sagacity  and  argument. 

And  not  less  learning,  penetration  and  masterly  reasoning  are 
conspicuous  in  his  HISTORICAL  ACCOUNT  OF  Two  NOTABLE 
CORRUPTIONS  OF  SCRIPTURES  IN  A  LETTER  TO  A  FRIEND.  This 


LIFE    OF    SIR    ISAAC    NEWTON.  41 

Treatise,  first  accurately  published  in  Dr.  Horsley's  edition  of  his 
works,  relates  to  two  texts :  the  one,  1  Epistle  of  St.  John  v.  7 ; 
the  other,  1  Epistle  of  St.  Paul  to  Timothy  iii.  16.  As  this 
work  had  the  effect  to  deprive  the  advocates  of  the  doctrine  of 
the  Trinity  of  two  leading  texts,  Newton  has  been  looked  upon 
as  an  Arian  ;  but  there  is  absolutely  nothing  in  his  writings  to 
warrant  such  a  conclusion. 

His  regaining  theological  works  consist  of  the  LEXICON  PRO- 
PHETICUM,  which  was  left  incomplete  ;  a  Latin  Dissertation  on 
the  sacred  cubit  of  the  Jews,  which  was  translated  into  English, 
and  published,  in  1737.  among  the  Miscellaneous  Works  of  John 
Greaves  ;  and  FOUR  LETTERS  addressed  to  Dr.  Bentlty,  contain 
ing  some  arguments  in  proof  of  a  Deity.  These  Letters  were 
dated  respectively  :  10th  December,  1692  ;  17th  January,  1693 ; 
25th  February,  1693;  and  llth  February,  1693— the  fourth 
bearing  an  earlier  date  than  the  third.  The  best  faculties  and 
the  profoundest  acquirements  of  our  author  are  convincingly 
manifest  in  these  lucid  and  powerful  compositions.  They  were 
published  in  1756,  and  reviewed  by  Dr.  Samuel  Johnson. 

Newton's  religious  writings  are  distinguished  by  their  absolute 
freedom  from  prejudice.  Everywhere,  throughout  them,  there 
glows  the  genuine  nobleness  of  soul.  To  his  whole  life,  indeed, 
we  may  here  fitly  extend  the  same  observation.  He  was  most 
richly  imbued  with  the  very  spirit  of  the  Scriptures  which  he  so 
delighted  to  study  and  to  meditate  upon.  His  was  a  piety,  so 
fervent,  so  sincere  and  practical,  that  it  rose  up  like  a  holy  incense 
from  every  thought  and  act.  His  a  benevolence  that  not  only 
willed,  but  endeavoured  the  best  for  all.  His  a  philanthropy 
that  held  in  the  embracings  of  its  love  every  brother-man. 
His  a  toleration  of  the  largest  and  the  truest ;  condemning  per 
secution  in  every,  even  its  mildest  form ;  and  kindly  encouraging 
each  striving  after  excellence  : — .1  toleration  that  came  not  of 
indifference — for  the  immoral  and  the  impious  met  with  their 
quick  rebuke — but  a  toleration  that  came  of  the  wise  humbleness 
and  the  Christian  charity,  which  see,  in  the  nothingness  of  self 
and  the  almightiness  of  TRUTH,  no  praise  for  the  ablest,  and  no 
blame  for  th^  feeblest  in  their  strugglings  upward  to  light  and  life. 


42  LIFE    OF    SIR    ISAAC    NEWTON, 

Tn  the  winter  of  1691-2,  on  returning  from  chapel,  one  morn 
ing,  Newton  foima  tnat  a  favourite  little  dog,  called  Diamond, 
had  overturned  a  lighted  taper  on  his  desk,  and  that  several  pa 
pers  containing  the  results  of  certain  optical  experiments,  were 
nearly  consumed.  His  only  exclamation,  on  perceiving  his  loss, 
was,  "  Oh  Diamond,  Diamond,  little  knowest  thou  the  mischiel 
thou  hast  done,"  Dr.  Brewster,  in  his  life  of  our  author,  gives  the 
following  extract  from  the  manuscript  Diary  of  Mr.  Abraham  De 
La  Pryme.  a  student  in  the  University  at  the  time  of  this  oc 
currence. 

"  1692.  February,  3. — What  I  heard  to-day  I  must  relate. 
There  is  one  Mr.  Newton  (whom  I  have  very  oft  seen),  Fellow 
of  Trinity  College,  that  is  mighty  famous  for  his  learning,  being  a 
most  excellent  mathematician,  philosopher,  divine,  &c.  He  has 
been  Fellow  of  the  Royal  Society  these  many  years  ;  and  among 
other  very  learned  books  and  tracts,  he:s  written  one  upon  the  mathe 
matical  principles  of  philosophy,  which  has  given  him  a  mighty 
name,  he  having  received,  especially  from  Scotland,  abundance  of 
congratulatory  letters  for  the  same ;  but  of  all  the  books  he  ever 
wrote,  there  was  one  of  colours  and  light,  established  upon  thou 
sands  of  experiments  which  he  had  been  twenty  years  of  making, 
and  which  had  cost  him  many  hundreds  of  pounds.  This  book 
which  he  vaiued  so  much,  and  which  was  so  much  talked  of,  had 
the  ill  luck  to  perish,  and  be  utterly  lost  just  when  the  learned 
author  was  almost  at  pitting  a  conclusion  at  the  same,  after  this 
manner :  In  a  winter's  morning,  leaving  it  among  his  other  papers 
on  his  study  table  while  he  went  to  chapel,  the  candle,  which  he 
had  unfortunately  left  burning  there,  too,  catched  hold  by  some 
means  of  other  papers,  and  they  fired  the  aforesaid  book,  and  ut 
terly  consumed  it  and  several  other  valuable  writings  ;  arid  which 
is  most  wonderful  did  no  further  mischief.  But  when  Mr.  New 
ton  came  from  chapel,  and  had  seen  what  was  done,  every  one 
thought  he  would  have  run  mad,  he  was  so  troubled  thereat  that 
he  was  not  himself  for  a  month  after.  A  long  account  of  this  his 
system  of  colours  you  may  find  in  the  Transactions  of  the  Royal 
Society,  which  he  had  sent  up  to  them  long  before  this  sad  mis 
chance  happened  unto  him." 


LIFE    OF    SIR    ISAAC    NEWTON.  43 

It  will  be  borne  in  mind  that  all  of  Newton's  theological  wri 
tings,  with  the  exception  of  the  Letters  to  Dr.  Bentley,  were 
composed  before  this  event  which,  we  must  conclude,  from 
Pryme's  words,  produced  a  serious  impression  upon  our  author  for 
about  a  month.  But  M.  Biot,  in  his  Life  of  Newton,  relying  on  a 
memorandum  contained  in  a  small  manuscript  Journal  of  Huygens, 
declares  this  occurrence  to  have  caused  a  deran-gement  of  New 
ton's  intellect.  M.  Blot's  opinions  and  deductions,  however,  as 
well  as  those  of  La  Place,  upon  this  subject,  were  based  upon 
erroneous  data,  and  have  been  overthrown  by  the  clearest  proof. 
There  is  not,  in  fact,  the  least  evidence  that  Newton's  reason  was, 
for  a  single  moment,  dethroned  ;  on  the  contrary,  the  testimony 
is  conclusive  that  he  was,  at  all  times,  perfectly  capable  of  carry 
ing  on  his  mathematical,  metaphysical  and  astronomical  inquiries. 
Loss  of  sleep,  loss  of  appetite,  and  irritated  nerves  will  disturb 
somewhat  the  equanimity  of  the  most  serene  ;  and  an  act  done,  or 
language  employed,  under  such  temporary  discomposure,  is  not  a 
just  criterion  of  the  general  tone  and  strength  of  a  man's  mind. 
As  to  the  accident  itself,  we  may  suppose,  whatever  might  have 
been  its  precise  nature,  that  it  greatly  distressed  him,  and,  still 
further,  that  its  shock  may  have  originated  the  train  of  nervous 
derangements,  which  afflicted  him,  more  or  less,  for  two  years 
afterward.  Yet,  during  this  very  period  of  ill  health,  we  find  him 
putting  forth  his  highest  powers.  In  1692,  he  prepared  for,  and 
transmitted  to  Dr.  Wallis  the  first  proposition  of  the  Treatise  on 
Quadratures,  with  examples  of  it  in  first,  second  and  third  flux 
ions.  He  investigated,  in  the  same  year,  the  subject  of  haloes ; 
making  and  recording  numerous  and  important  observations  rela 
tive  thereto.  Those  profound  and  beautiful  Letters  to  Dr.  Bentley 
were  written  at  the  close  of  this  and  the  beginning  of  the  next 
year.  In  October,  1693,  Locke,  who  was  then  about  publishing  a 
second  edition  of  his  work  on  the  Human  Understanding,  request 
ed  Newton  to  reconsider  his  opinions  on  innate  ideas.  And  in 
1694,  he  was  zealously  occupied  in  perfecting  his  lunar  theory  ; 
visiting  Flamstead,  at  the  Royal  Observatory  of  Greenwich,  in 
September,  and  obtaining  a  series  of  lunar  observations  ;  and 


14  LIFE    OF    SIR    ISAAC    NEWTON. 

commencing,  in  October,  a  correspondence  with  that  distinguished 
practical  Astronomer,  which  continued  till  1698. 

We  now  arrive  at  the  period  when  Newton  permanently  with 
drew  from  the  seclusion  of  a  collegiate,  and  entered  upon  a  more 
active  and  public  life.  He  was  appointed  Warden  of  the  Mint, 
in  1695,  through  the  influence  of  Charles  Montague,  Chancellor 
of  the  Exchequer,  and  afterward  Earl  of  Halifax.  The  current 
roin  of  the  nation  had  been  adulterated  and  debased,  and  Mon 
tague  undertook  a  re-coinage.  Our  author's  mathematical  and 
chemical  knowledge  proved  eminently  useful  in  accomplishing 
this  difficult  and  most  salutary  reform.  In  1699,  he  was  pro 
moted  to  the  Mastership  of  the  Mint — an  office  worth  twelve  or 
fifteen  hundred  pounds  per  annum,  and  which  he  held  during  the 
remainder  of  his  life.  He  wrote,  in  this  capacity,  an  official  Re 
port  on  the  Coinage,  which  has  been  published ;  he  also  prepared 
a  Table  of  Assays  of  Foreign  Coins,  which  was  printed  at  the 
end  of  Dr.  Arbuthnot's  Tables  of  Ancient  Coins,  Weights,  and 
Measures,  in  1727. 

Newton  retained  his  Professorship  at  Cambridge  till  1703. 
But  he  had,  on  receiving  the  appointment  of  Master  of  the  Mint, 
in  1699,  made  Mr.  Whiston  his  deputy,  with  all  the  emoluments 
of  the  office  ;  and,  on  finally  resigning,  procured  his  nomination  to 
the  vacant  Chair. 

In  January  1697,  John  Bernouilli  proposed  to  the  most  distin 
guished  mathematicians  of  Europe  two  problems  for  solution. 
Leibnitz,  admiring  the  beauty  of  one  of  them,  requested  the  time 
for  solving  it  to  be  extended  to  twelve  months — twice  the  period 
originally  named.  The  delay  was  readily  granted.  Newton,  how 
ever,  sent  in,  the  day  after  he  received  the  problems,  a  solution  of 
them  to  the  President  of  the  Royal  Society.  Bernouilli  obtained 
solutions  from  Newton,  Leibinitz  and  the  Marquis  De  L'Hopital ; 
but  Newton's  though  anonymous,  he  immediately  recognised 
" tanquam  ungue  leonem"  as  the  lion  is  known  by  his  claw. 
We  may  mention  here  the  famous  problem  of  the  trajectories 
proposed  by  Leibnitz,  in  1716,  for  the  purpose  of  "feeling  the 
pulse  of  the  English  Analysts."  Newton  received  the  problem 
about  five  o'clock  in  the  afternoon,  as  he  was  returning  from  the 


LIFE    OF    SIR    ISAAC    NEWTON.  45 

Mint ;  and  though  it  was  extremely  difficult  and  he  himself  much 
fatigued,  yet  he  completed  its  solution,  the  same  evening  before 
he  went  to  bed. 

The  history  of  these  problems  affords,  by  direct  comparison,  a 
striking  illustration  of  Newton's  vast  superiority  of  mind.  That 
amazing  concentration  and  grasp  of  intellect,  of  which  we  have 
spoken,  enabled  him  to  master  speedily,  and,  as  it  were,  by  a 
single  effort,  those  things,  for  the  achievement  of  which,  the  many 
would  essay  utterly  in  vain,  and  the  very,  very  few  attain  only 
after  long  and  renewed  striving.  And  yet,  with  a  modesty  as 
unparalleled  as  his  power,  he  attributed  his  successes,  not  to  any 
extraordinary  sagacity,  but  solely  to  industry  and  patient  thought. 
Mr-  kept  the  subject  of  consideration  constantly  before  him,  and 
waited  till  the  first  dawning  opened  gradually  into  a  full  and 
clear  light ;  never  quitting,  if  possible,  the  mental  process  till  the 
object  of  it  were  wholly  gained.  He  never  allowed  this  habit  of 
meditation  to  appear  in  his  intercourse  with  society ;  but  in  the 
privacy  of  his  own  chamber,  or  in  the  midst  of  his  own  family,  he 
gave  himself  up  to  the  deepest  abstraction.  Occupied  with  some 
interesting  investigation,  he  would  often  sit  down  on  his  bedside, 
after  he  rose,  and  remain  there,  for  hours,  partially  dressed. 
Meal-time  would  frequently  come  and  pass  unheeded ;  so  that, 
unless  urgently  reminded,  he  would  neglect  to  take  the  re 
quisite  quantity  of  nourishment.  But  notwithstanding  his  anx 
iety  to  be  left  undisturbed,  he  would,  when  occasion  required, 
turn  aside  his  thoughts,  though  bent  upon  the  most  intricate  re 
search,  and  then,  when  leisure  served,  again  direct  them  to  the 
very  point  where  they  ceased  to  act :  and  this  he  seemed  to  ac 
complish  not  so  much  by  the  force  of  his  memory,  as  by  the  force 
of  his  inventive  faculty,  before  the  vigorous  intensity  of  which,  no 
subject,  however  abstruse,  remained  long  unexplored. 

Me  was  elected  a  member  of  the  Royal  Academy  of  Sciences 
at  Paris,  in  1699,  when  that  distinguished  Body  were  empowered, 
by  a  new  charter,  to  admit  a  small  number  of  foreign  associates. 
In  1700,  he  communicated  to  Dr.  Halley  a  description  of  his  re 
flecting  instrument  for  observing  the  moon's  distance  from  the 
fixed  stars.  This  description  was  published  in  the  Philosophical 


46  LIFE    OF    SIR    ISAAC    NEWTON, 

Transactions,  in  1742.  The  instrument  was  the  same  as  that 
produced  by  Mr.  Hadley,  in  1731,  and  which,  under  the  name  of 
Hadley's  Quadrant,  has  been  of  so  great  use  in  navigation.  On 
the  assembling  of  the  new  Parliament,  in  1701,  Newton  was  re- 
elected  one  of  the  members  for  the  University  of  Cambridge.  In 
1703,  he  was  chosen  President  of  the  Royal  Society  of  London, 
to  which  office  he  was  annually  re-elected  till  the  period  of  his 
decease — about  twenty-five  years  afterward. 

Our  author  unquestionably  devoted  more  labour  to,  and,  in 
many  respects,  took  a  greater  pride  in  his  Optical,  than  his  other 
discoveries.  This  science  he  had  placed  on  a  new  and  indestruc 
tible  basis ;  and  he  wished  not  only  to  build,  but  to  perfect  the 
costly  and  glowing  structure.  He  had  communicated,  before  the 
publication  of  the  PRINCIPIA,  his  most  important  researches  on 
light  to  the  Royal  Society,  in  detached  papers  which  were  inserted 
in  successive  numbers  of  the  Transactions ;  but  he  did  not  pub 
lish  a  connected  view  of  these  labours  till  1704,  when  they  appeared 
under  the  title  of  OPTICS  :  OR,  A  TREATISE  ON  THE  REFLEXIONS, 
REFRACTIONS,  INFLEXIONS  AND  COLOURS  OF  LIGHT.  To  this, 
but  to  no  subsequent  edition,  were  added  two  Mathematical  Trea 
tises,  entitled,  TRACTATUS  DUO  DE  SPECIEBUS  ET  MAGNITUDINE 
FIGURARUM  cuRViLiNEARUM ;  the  one  bearing  the  title  TRACTATUS 
DE  QUADRATURA  CuRVARUM ;  and  the  other,  that  of  ENUMERATIO 
LINEARUM  TERTII  ORDiNis.  The  publication  of  these  Mathemati 
cal  Treatises  was  made  necessary  in  consequence  of  plagiarisms 
from  the  manuscripts  of  them  loaned  by  the  author  to  his  friends. 
Dr.  Samuel  Clarke  published  a  Latin  translation  of  the  Optics,  in 
in  1706  ;  whereupon  he  was  presented  by  Newton,  as  a  mark  of 
his  grateful  approbation,  with  five  hundred  pounds,  or  one  hun 
dred  pounds  for  each  of  his  children.  The  work  was  afterward 
translated  into  French.  It  had  a  remarkably  wide  circulation, 
and  appeared,  in  several  successive  editions,  both  in  England  and 
on  the  Continent.  There  is  displayed,  particularly  on  this  Opti 
cal  Treatise,  the  author's  talent  for  simplifying  and  communica 
ting  the  profoundest  speculations.  It  is  a  faculty  rarely  united  to 
that  of  the  highest  invention.  Newton  possessed  both  ;  and  thus 
that  mental  perfectness  which  enabled  him  to  create,  to  combine, 


LIFE    OF    SIR    ISAAC    NEWTON.  47 

and  to  teach,  and  so  render  himself,  not  the  "ornament"  cnly; 
but  inconceivably  more,  the  pre-eminent  benefactor  of  his  species. 

The  honour  of  knighthood  v/as  conferred  on  our  author  in 
1705.  Soon  afterward,  he  was  a  candidate  again  for  the  Repre 
sentation  of  the  University,  but  was  defeated  by  a  large  majority. 
It  is  thought  that  a  more  pliant  man  was  preferred  by  both  min 
isters  and  electors.  Newton  was  always  remarkable  for  simplicity 
of  dress,  and  his  only  known  departure  from  it  was  on  this  oc 
casion,  when  he  is  said  to  have  appeared  in  a  suit  of  laced 
clothes. 

The  Algebraical  Lectures  which  he  had,  Juring  nine  years, 
delivered  at  Cambridge,  were  published  by  Whiston,  in  1707, 
under  the  title  of  ARITHMETICS  UNIVERSALIS,  SINE  DE  COMPOSI 
TIONS  ET  RESOLUTIONS  ARITHMETICA  LIBER.  This  publication 
is  said  to  have  been  a  breach  of  confidence  on  Whiston's  part.  Mr. 
Ralphson,  not  long  afterward,  translated  the  work  into  English ; 
and  a  second  edition  of  it,  with  improvements  by  the  author,  was 
issued  at  London,  1712,  by  Dr.  Machin.  Subsequent  editions, 
both  in  English  and  Latin,  with  commentaries,  have  been  published. 

In  June,  1709,  Newton  intrusted  the  superintendence  of  a 
second  edition  of  the  PRINCIPIA  to  Roger  Cotes,  Plumian  Pro 
fessor  of  Astronomy  at  Cambridge.  The  first  edition  had  been 
sold  off  for  some  time.  Copies  of  the  work  had  become  very 
rare,  and  could  only  be  obtained  at  several  times  their  original 
cost.  A  great  number  of  letters  passed  oetween  the  author  and 
Mr.  Cotes  during  the  preparation  of  the  edition,  which  finally 
appeared  in  May,  1713.  It  had  many  alterations  and  improve 
ments,  and  was  accompanied  by  an  admirable  Preface  from  the 
pen  of  Cotes. 

Our  author's  early  Treatise,  entitled,  ANALYSIS  PER  EQUATIONES 
NUMERO  TERMINORUM  INFINITAS,  as  well  as  a  small  Tract,  Gearing 
the  title  of  METHODUS  DIFFERENTIALS,  was  published,  witn  nis 
consent,  in  1711.  The'  former  of  these,  and  the  Treatise  De 
Quadratura  Curvarum,  translated  into  Englisn,  witn  a  .arge  com 
mentary,  appeared  in  1745.  His  work,  entitled.  ARTIS  ANA 
LYTICS  SPECIMINA,  VEL  GEOMETRIA  ANALYTICA,  was  iirs;  given 
to  the  world  in  the  edition  of  Dr.  Horsley,  1779. 


48  LIFE    OF    SIR    ISAAC    NEWTON. 

It  is  a  notable  fact,  in  Newton's  history,  that  he  never  volun* 
tarily  published  any  one  of  his  purely  mathematical  writings 
The  cause  of  this  unwillingness  in  some,  and,  in  other  instances, 
of  his  indifference,  or,  at  least,  want  of  solicitude  to  put  forth  his 
works  may  be  confidently  sought  for  in  his  repugnance  to  every 
thing  like  contest  or  dispute.  But,  going  deeper  than  this  aver 
sion,  we  find,  underlying  his  whole  character  and  running  parallel 
with  all  his  discoveries,  that  extraordinary  humility  which  always 
preserved  him  in  a  position  so  relatively  just  to  the  behests  of 
time  and  eternity,  that  the  infinite  value  of  truth,  and  the  utter 
worthlessness  of  fame,  were  alike  constantly  present  to  him. 
Judging  of  his  course,  however,  in  its  more  temporary  aspect,  as 
bearing  upon  his  immediate  quiet,  it  seemed  the  most  unfortunate. 
For  an  early  publication,  especially  in  the  case  of  his  Method  of 
Fluxions,  would  have  anticipated  all  rivalry,  and  secured  him 
from  the  contentious  claims  of  Leibnitz.  Still  each  one  will  solve 
the  problem  of  his  existence  in  his  own  way,  and,  with  a  manlike 
Newton,  his  own,  as  we  conceive,  could  be  no  other  than  the  best 
way.  The  conduct  of  Leibnitz  in  this  affair  is  quite  irreconcilable 
with  the  stature  and  strength  of  the  man ;  giant-like,  and  doing 
nobly,  in  many  ways,  a  giant's  work,  yet  cringing  himself  into  the 
dimensions  and  performances  of  a  common  calumniator.  Opening 
in  1699,  the  discussion  in  question  continued  till  the  close  of 
Leibnitz's  life,  in  1716.  We  give  the  summary  of  the  case  as 
contained  in  the  Report  of  the  Committee  of  the  Royal  Society, 
the  deliberately  weighed  opinion  of  which  has  been  adopted  as  an 
authoritative  decision  in  all  countries. 

"  We  have  consulted  the  letters  and  letter  books  in  the  custody 
of  the  Royal  Society,  and  those  found  among  the  papers  of  Mr. 
John  Collins,  dated  between  the  years  1669  and  1677,  inclusive  ; 
and  showed  them  to  such  as  knew  and  avouched  the  hands  of  Mr. 
Barrow,  Mr.  Collins,  Mr.  Oldenburg,  and  Mr.  Leibnitz ;  and 
compared  those  of  Mr.  Gregory  with  one  another,  and  with  copies 
of  some  of  them  taken  in  the  hand  of  Mr.  Collins  ;  and  have 
extracted  from  them  what  relates  to  the  matter  referred  to  us : 
all  which  extracts,  herewith  delivered  to  you,  we  believe  to  be 
genuine  and  authentic.  And  by  these  letters  and  papers  wf 
find:  — 


LIFE    OF    SIR    ISAAC    NEWTON.  49 

"  I.  Mr.  Leibnitz  was  in  London  in  the  beginning  of  the  year 
1673  ;  and  went  thence  in  or  about  March,  to  Paris,  where  he 
kept  a  correspondence  with  Mr.  Collins,  by  means  of  Mr.  Olden 
burg,  till  about  September,  1676,  and  then  returned,  by  London 
and  Amsterdam,  to  Hanover:  and  that  Mr.  Collins  was  very  free 
in  communicating  to  able  mathematicians  what  he  had  received 
from  Mr,  Newton  and  Mr.  Gregory. 

"  II.  That  when  Mr.  Leibnitz  was  the  first  time  in  London, 
he  contended  for  the  invention  of  another  differential  method, 
properly  so  called  ;  and,  notwithstanding  he  was  shown  by  Dr. 
Pell  that  it  was  Newton?s  method,  persisted  in  maintaining  it  to 
be  his  own  invention,  by  reason  that  he  had  found  it  by  himself 
without  knowing  what  Newton  had  done  before,  and  had  much 
improved  it.  And  we  find  no  mention  of  his  having  any  other 
differential  method  than  Newton's  before  his  letter  of  the  21st  of 
June,  1677,  which  was  a  year  after  a  copy  of  Mr.  Newton's  letter 
of  the  10th  of  December,  1672,  had  been  sent  to  Paris  to  be 
communicated  to  him ;  and  above  four  years  after  Mr.  Collins 
began  to  communicate  that  letter  to  his  correspondents  ;  in  which 
letter  the  method  of  fluxions  was  sufficiently  described  to  any 
intelligent  person. 

"III.  That  by  Mr.  Newton's  letter,  of  the  13th  of  June,  1676 
it  appears  that  he  had  the  method  of  fluxions  above  five  years 
before  the  writing  of  that  letter.  And  by  his  Analysis  per  ^Equa- 
tiones  numero  Terminorum  Infmitas,  communicated  by  Dr.  Barrow 
to  Mr.  Collins,  in  July,  1669,  we  find  that  he  had  invented  the 
method  before  that  time. 

"IV.  That  the  differential  method  is  one  and  the  same  with 
the  method  of  fluxions,  excepting  the  name  and  mode  of  notation  ; 
Mr.  Leibnitz  calling  those  quantities  differences  which  Mr.  Newton 
calls  moments,  or  fluxions ;  and  marking  them  with  a  letter  d — a 
mark  not  used  by  Mr.  Newton. 

"  And,  therefore,  we  take  the  proper  question  to  be,  not  who 
invented  this  or  that  method,  but,  who  was  the  first  inventor  of 
the  method  ?  And  we  believe  that  those  who  have  reputed  Mr. 
Leibnitz  the  first  inventor  knew  little  or  nothing  of  his  correspond 
ence  with  Mr.  Collins  and  Mr.  Oldenburg  long  before,  nor  of  Mr. 


50  LIFE    OP    SIR    ISAAC    NEWTON. 

Newton's  hiving  that   method   above   fifteen  years  before   Mr 
Leibnitz  began  to  publish  it  in  the  Acta  Eruditorum  of  Leipsic. 

"  For  which  reason  we  reckon  Mr.  Newton  the  first  inventor ; 
and  are  of  opinion  that  Mr.  Keill,  in  asserting  the  same,  has  been 
no  ways  injurious  to  Mr.  Leibnitz.  And  we  submit  to  the  judg 
ment  of  the  Society,  whether  the  extract  and  papers,  now  pre 
sented  to  you,  together  with  what  is  extant,  to  the  same  pur 
pose,  in  Dr.  Wallis's  third  volume,  may  not  deserve  to  be  made 
public." 

This  Report,  with  the  collection  of  letters  and  manuscripts, 
under  the  title  of  COMMERCIUM  EPISTOLICUM  D.  JOHANNIS  COLLINS 

ET     ALIORUM     DE    ANALYSI    PROMOTA     JuSSU     SoCIETATIS     REGIES 

EDITUM,  appeared  accordingly  in  the  early  part  of  1713.  Its 
publication  seemed  to  infuse  additional  bitterness  into  the  feelings 
of  Leibnitz,  who  descended  to  unfounded  charges  and  empty 
threats.  He  had  been  privy  counsellor  to  the  Elector  of  Han 
over,  before  that  prince  was  elevated  to  the  British  throne ;  and 
in  his  correspondence,  in  1715  and  1716,  with  the  Abbe  Conti, 
then  at  the  court  of  George  L,  and  with  Caroline,  Princess  of 
Wales,  he  attacked  the  doctrines  of  the  PRINCIPIA,  and  indirectly 
its  author,  in  a  manner  very  discreditable  to  himself,  both  as  a 
learned  and  as  an  honourable  man.  His  assaults,  however,  were 
triumphantly  met;  and,  to  the  complete  overthrow  of  his  rival 
pretensions,  Newton  was  induced  to  give  the  finishing  blow.  The 
verdict  is  universal  and  irreversible  that  the  English  preceded 
the  German  philosopher,  by  at  least  ten  years,  in  the  invention 
of  fluxions.  Newton  could  not  have  borrowed  from  Leibnitz ; 
but  Leibnitz  might  have  borrowed  from  Newton.  A  new  edition 
of  the  Commercium  Epistolicum  was  published  in  1722-5  (?)  ;  but 
neither  in  this,  nor  in  the  former  edition,  did  our  author  take  any 
part.  The  disciples,  enthusiastic,  capable  and  ready,  effectually 
shielded,  with  the  buckler  of  Truth,  the  character  of  the  Master, 
whose  own  conduct  throughout  was  replete  with  delicacy,  dignity 
and  justice.  He  kept  aloof  from  the  controversy — in  which  Dr. 
Keill  stood  forth  as  the  chief  representative  of  the  Newtonian 
side — till  the  very  last,  when,  for  the  satisfaction  of  the  King, 
George  L.  rather  than  for  his  own,  he  consented  to  put  forth  his 


LIFE    OF    SI|        L^.-vJ    NEWTON.  5i 

hand  and  firmly  secure  his  rights  upon  a  certain  and  impregnable, 
basis. 

A  petition  to  have  inventions  for  promoting  the  discovery  of  the 
longitude  at  sea,  suitably  rewarded,  was  presented  to  the  House 
of  Commons,  in  1714.  A  committee,  having  been  appointed  to 
investigate  the  subject,  called  upon  Newton  and  others  for  their 
opinions.  That  of  our  author  was  given  in  writing,  A  report, 
favourable  to  the  desired  measure,  was  then  taken  up,  and  a  bill 
for  its  adoption  subsequently  passed. 

On  the  ascension  of  George  I.,  in  1714,  Newton  became  an 
object  of  profound  interest  at  court.  His  position  under  govern 
ment,  his  surpassing  fame,  his  spotless  character,  and.  above  all, 
his  deep  and  consistent  piety,  attracted  the  reverent  regard  of  the 
Princess  of  Wales,  afterward  queen -consort  to  George  II.  She 
was  a  woman  of  a  highly  cultivated  mind,  and  derived  the  greatest 
pleasure  from  conversing  with  Newton  and  corresponding  with 
Leibnitz.  One  day,  in  conversation  with  her,  our  author  men 
tioned  and  explained  a  new  system  of  chronology,  which  he  had 
composed  at  Cambridge,  where  he  had  been  in  the  habit  "  of 
refreshing  himself  with  history  and  chronology,  when  he  wac 
weary  with  other  studies."  Subsequently,  in  the  year  1718,  she 
requested  a  copy  of  this  interesting  and  ingenious  work  Newton, 
accordingly,  drew  up  an  abstract  of  the  system  from  the  separate 
papers  in  which  it  existed,  and  gave  it  to  her  on  condition  that  it 
should  riot  be  communicated  to  any  other  person.  Sometime 
afterward  she  requested  that  the  Abbe  Conti  might  be  allowed 
to  have  a  copy  of  it  The  author  consented:  and  the  abbe 
received  a  copy  of  the  manuscript,  under  the  like  injunction  and 
promise  of  secrecy.  This  manuscript  bore  the  title  of  "  A  short 
Chronicle,  from  the  First  Memory  of  Tilings  in  Europe,  to  the 
Conquest  of  Persia,  by  Alexander  the  Great." 

After  Newton  took  up  his  residence  in  London,  he  lived  in  a 
style  suited  to  his  elevated  position  and  rank.  He  kept  his  car 
riage,  with  an  establishment  of  three  male  and  three  female  serv 
ants.  But  to  everything  like  vain  show  and  luxury  he  was  utterly 
averse.  His  household  affairs,  for  the  last  twenty  years  of  his 
life,  were  under  the  charge  of  his  niece,  Mrs.  Catherine  Barton, 


52  LIFE    OF    SIR    ISAAC    NEWTON. 

wife  and  widow  of  Colonel  Barton — a  woman  of  great  beauty  and 
accomplishment — and  subsequently  married  to  John  Conduit,  Esq. 
At  home  Newton  was  distinguished  by  that  dignified  and  gentle 
hospitality  which  springs  alone  from  true  nobleness.  On  all  pro 
per  occasions,  he  gave  splendid  entertainments,  though  without 
ostentation.  In  society,  whether  of  the  palace  or  the  cottage, 
his  manner  was  self-possessed  and  urbane ;  his  look  benign  and 
affable  ;  his  speech  candid  and  modest ;  his  whole  air  undisturb 
edly  serene.  He  had  none  of  what  are  usually  called  the  singu 
larities  of  genius  ;  suiting  himself  easily  to  every  company — 
except  that  of  the  vicious  and  wicked ;  and  speaking  of  himself 
and  others,  naturally,  so  as  never  even  to  be  suspected  of  vanity. 
There  was  in  him,  if  we  may  be  allowed  the  expression,  a  WHOLE 
NESS  of  nature,  which  did  not  admit  of  such  imperfections  and 
weakness — the  circle  was  too  perfect,  the  law  too  constant,  and 
the  disturbing  forces  too  slight  to  suffer  scarcely  any  of  those 
eccentricities  which  so  interrupt  and  mar  the  movements  of  many 
bright  spirits,  rendering  their  course  through  the  world  more  like 
that  of  the  blazing  meteor  than  that  of  the  light  and  life-impart 
ing  sun.  In  brief,  the  words  GREATNESS  and  GOODNESS  could 
not,  humanly  speaking,  be  more  fitly  employed  than  when  applied 
as  the  pre-eminent  characteristics  of  this  pure,  meek  and  vene 
rable  sage. 

In  the  eightieth  year  of  his  age,  Newton  was  seized  with 
symptoms  of  stone  in  the  bladder.  His  disease  was  pronounced 
incurable.  He  succeeded,  however,  by  means  of  a  strict  regimen, 
and  other  precautions,  in  alleviating  his  complaint,  and  procuring 
long  intervals  of  ease.  His  diet,  always  frugai,  was  now  extremely 
temperate,  consisting  chiefly  of  broth,  vegetables,  and  fruit,  with, 
now  and  then,  a  little  butcher  meat.  He  gave  up  the  use  of  his 
carriage,  and  employed,  in  its  stead,  when  he  went  out,  a  chair. 
All  invitations  to  dinner  were  declined ;  and  only  small  parties 
were  received,  occasionally,  at  his  own  house. 

In  1724  he  wrote  to  the  Lord  Provost  of  Edinburgh,  offering 
to  contribute  twenty  pounds  yearly  toward  the  salary  of  Mr. 
Maclaurin,  provided  he  accepted  the  assistant  Professorship  of 
Mathematics  in  the  University  of  that  place.  Not  only  in  the 


LIFE    OP    SIR    ISAAC    NEWTON.  53 

cause  of  ingenuity  and  learning,  but  in  that  of  religion — in  relieving 
the  poor  and  .assisting  his  relations,  Newton  annually  expended 
large  sums.  He  was  generous  and  charitable  almost  to  a  fault. 
Those,  he  would  often  remark,  who  gave  away  nothing  till  they 
died,  never  gave  at  all.  His  wealth  had  become  considerable  by 
a  prudent  economy ;  but  he  regarded  money  in  no  other  light 
than  as  one  of  the  means  wherewith  he  had  been  intrusted  to  do 
good,  and  he  faithfully  employed  it  accordingly. 

He  experienced,  in  spite  of  all  his  precautionary  measures,  a 
return  of  his  complaint  in  the  month  of  August,  of  the  same  year, 
1 724,  when  he  passed  a  stone  the  size  of  pea  ;  it  came  from  him 
in  two  pieces,  the  one  at  the  distance  of  two  day.s  from  the  other. 
Tolerable  good  health  then  followed  for  some  months.  In  Janu 
ary,  1725,  however,  he  was  taken  with  a  violent  cough  and  inflam 
mation  of  the  lungs.  In  consequence  of  this  attack,  he  was  pre 
vailed  upon  to  remove  to  Kensington,  where  his  health  greatly 
improved.  In  February  following,  he  was  attacked  in  both  feet 
with  the  gout,  of  the  approach  of  which  he  had  received,  a  few 
years  before,  a  slight  warning,  and  the  presence  of  which  now 
produced  a  very  beneficial  change  in  his  general  health.  Mr. 
Conduit,  his  nephew,  has  recorded  a  curious  conversation  which 
took  place,  at  or  near  this  time,  between  himself  and  Sir  Isaac. 

"I  was,  on  Sunday  night,  the  7th  March,  1724-5,  at  Kensing 
ton,  with  Sir  Isaac  Newton,  in  his  lodgings,  just  after  he  was  out 
of  a  fit  of  the  gout,  which  he  had  had  in  both  of  his  feet,  for  the 
first  time,  in  the  eighty-third  year  of  his  age.  He  was  better  after 
it,  and  his  head  clearer  and  memory  stronger  than  I  had  known 
them  for  some  time.  He  then  repeated  to  me,  by  way  of  dis 
course,  very  distinctly,  though  rather  in  answer  to  my  queries, 
than  in  one  continued  narration,  what  he  had  often  hinted  to  me 
before,  viz. :  that  it  was  his  conjecture  (he  would  affirm  nothing) 
that  there  was  a  sort  of  revolution  in  the  heavenly  bodies  ;  that 
the  vapours  and  light,  emitted  by  the  sun,  which  had  their  sedi 
ment,  as  water  and  other  matter,  had  gathered  themselves,  by 
degrees,  into  a  body,  and  attracted  more  matter  from  the  planets, 
and  at  last  made  a  secondary  planet  (viz. :  one  of  those  that  go 
round  another  planet),  and  then,  by  gathering  to  them,  and 


54  LIFE    OF    SIR    ISAAC    NEWTON. 

attracting  more  matter,  became  a  primary  planet ;  and  then,  bf 
increasing  still,  became  a  comet,  which,  after  certain  revolutions, 
by  coming  nearer  and  nearer  to  the  sun,  had  all  its  volatile  parts 
condensed,  and  became  a  matter  tit  to  recruit  and  replenish  the 
sun  (which  must  waste  by  the  constant  heat  and  light  it  emitted), 
as  a  faggot  would  this  fire  if  put  into  it  (we  were  sitting  by  a 
wood  fire),  and  that  that  would  probably  be  the  effect  of  the 
comet  of  1680,  sooner  or  later  ;  for,  by  the  observations  made 
upon  it,  it  appeared,  before  it  came  near  the  sun,  with  a  tail  only 
two  or  three  degrees  long  ;  but,  by  the  heat  it  contracted,  in  going 
so  near  the  sun,  it  seemed  to  have  a  tail  of  thirty  or  forty  degrees 
when  it  went  frpm  it ;  that  he  could  not  say  when  this  comet 
would  drop  into  the  sun ;  it  might  perhaps  have  five  or  six  revo 
lutions  more  first,  but  whenever  it  did  it  would  so  much  increase 
the  heat  of  the  sun  that  this  earth  would  be  burned,  and  no  ani 
mals  in  it  could  live.  That  he  took  the  three  phenomena,  seen 
by  Hipparchus,  Tycho  Brahe,  and  Kepler's  disciples,  to  have  been 
of  this  kind,  for  he  could  not  otherwise  account  for  an  extraor 
dinary  light,  as  those  were,  appearing,  all  at  once,  among  the 
the  fixed  stars  (all  which  he  took  to  be  suns,  enlightening  other 
planets,  as  our  sun  does  ours),  as  big  as  Mercury  or  Venus  seems 
to  us,  and  gradually  diminishing,  for  sixteen  months,  and  then 
sinking  into  nothing.  He  seemed  to  doubt  whether  there  were 
not  intelligent  beings,  superior  to  us,  who  superintended  these 
revolutions  of  the  heavenly  bodies,  by  the  direction  of  the  Supreme 
Being.  He  appeared  also  to  be  very  clearly  of  opinion  that  the 
inhabitants  of  this  world  were  of  short  date,  and  alledged,  as  one 
reason  for  that  opinion,  that  all  arts,  as  letters,  ships,  printing, 
needle,  &c.,  were  discovered  within  the  memory  of  history,  which 
could  not  have  happened  if  the  world  had  been  eternal ;  and  that 
there  were  visible  marks  of  ruin  upon  it  which  could  not  be 
effected  by  flood  only.  When  I  asked  him  how  this  earth  could 
have  been  repeopled  if  ever  it  had  undergone  the  same  fate 
it  was  threatened  with  hereafter,  by  the  comet  of  1680,  he 
answered,  that  required  the  power  of  a  Creator.  He  said  he 
took  all  the  planets  to  be  composed  of  the  same  matter  with  this 
earth,  viz. :  earth,  water,  stones,  &c.3  but  variously  concocted.  J 


LIFE    OP    SIR    ISAAC    NEWTON.  55 

asked  him  why  he  would  not  publish  his  conjectures,  as  conjec 
tures,  and  instanced  that  Kepler  had  communicated  his ;  and 
though  he  had  not  gone  near  so  far  as  Kepler,  yet  Kepler's 
guesses  were  so  just  and  happy  that  they  had  been  proved  and 
demonstrated  by  him.  His  answer  was,  "  I  do  not  deal  in  con 
jectures."  But,  on  my  talking  to  him  about  the  four  observations 
that  had  been  made  of  the  comet  of  1680,  at  574  years'  distance, 
and  asking  him  the  particular  times,  he  opened  his  Principia, 
which  laid  on  the  table,  and  showed  me  the  particular  periods, 
viz.:  1st.  The  Julium  Sidus,  in  the  time  of  Justinian,  in  1106, 
in  1680. 

"  And  I,  observing  that  he  said  there  of  that  comet,  '  incidet 
in  corpus  solis,'  and  in  the  next  paragraph  adds,  '  stellae  fixae 
refici  possunt,'  told  him  I  thought  he  owned  there  what  we  had 
been  talking  about,  viz. :  that  the  comet  would  drop  into  the  sun, 
and  that  fixed  stars  were  recruited  and  replenished  by  comets 
when  they  dropped  into  them ;  and,  consequently,  that  the  sun 
would  be  recruited  too ;  and  asked  him  why  he  would  not  own  as 
fully  what  he  thought  of  the  sun  as  well  as  what  he  thought  of 
the  fixed  stars.  He  said,  'that  concerned  us  more;'  and,  laugh 
ing,  added,  that  he  had  said  enough  for  people  to  know  his 
meaning." 

In  the  summer  of  1725,  a  French  translation  of  the  chronolo 
gical  MS.,  of  which  the  Abbe  Conti  had  been  permitted,  some 
time  previous,  to  have  a  copy,  was  published  at  Paris,  in  violation 
of  all  good  faith.  The  Punic  Abbe  had  continued  true  to  his 
promise  of  secrecy  while  he  remained  in  England  ;  but  no  sooner 
did  he  reach  Paris  than  he  placed  the  manuscript  into  the  hands 
of  M.  Freret,  a  learned  antiquarian,  who  translated  the  work,  and 
accompanied  it  with  an  attempted  refutation  of  the  leading  points 
of  the  system.  In  November,  of  the  same  year,  Newton  received 
a  presentation  copy  of  this  publication,  which  bore  the  title  of 
ABREGE  DE  CHRONOLOGIE  DE  M.  LE  CHEVALIER  NEWTON,  FAIT 

PAR  LUI-MEME,  ET   TRADUIT    SUR    LE    MANUSCRIPT   ANGLAIS.        Soon 

afterward  a  paper  entitled,  REMARKS  ON  TFE  OBERVATIONS  MADE 
ON  A  CHRONOLOGICAL  INDEX  OF  SIR  ISAAC  NE.WTON,  TRANSLATED 
INTO  FRENCH  BY  THE  OBSERVATOR,  ANL  PUBLISHED  AT  PARIS, 


56  LIFE    OF    SIR    ISAAC    NEWTON, 

was  drawn  up  by  our  author,  and  printed  in  the  Philosophical 
Transactions  for  1725.  It  contained  a  history  of  the  whole 
matter,  and  a  triumphant  reply  to  the  objections  of  M.  Freret. 
This  answer  called  into  the  field  a  fresh  antagonist,  Father  Soueiet, 
whose  five  dissertations  on  this  subject  were  chiefly  remarkable 
for  the  want  of  knowledge  and  want  of  decorum,  which  they 
displayed.  In  consequence  of  these  discussions,  Newton  was  in 
duced  to  prepare  his  larger  work  for  the  press,  and  had  nearly 
completed  it  at  the  time  of  his  death.  It  was  published  in  1728, 
under  the  title  of  THE  CHRONOLOGY  OF  THE  ANCIENT  KINGDOMS 
AMENDED,  TO  WHICH  is  PREFIXED  A  SHORT  CHRONICLE  FROM  THE 

FIRST  MEMORY  OF  THINGS   IN    EUROPE    TO    THE    CONQUEST    OF 

PERSIA  BY  ALEXANDER  THE  GREAT.  It  consists  of  six  chap 
ters:  1.  On  the  Chronology  of  the  Greeks;  according  to  Whis- 
ton,  our  author  wrote  out  eighteen  copies  of  this  chapter  with  his 
own  hand,  differing  little  from  one  another.  2.  Of  the  Empire 
of  Egypt;  3.  Of  the  Assyrian  Empire;  4.  Of  the  two  contempo 
rary  Empires  of  the  Babylonians  and  Medes ;  5.  A  Description 
of  the  Temple  of  Solomon  ;  6.  Of  the  Empire  of  the  Persians  ; 
this  chapter  was  not  found  copied  with  the  other  five,  but  as  it 
was  discovered  among  his  papers,  arid  appeared  to  be  a  continu 
ation  of  the  same  work,  the  Editor  thought  proper  to  add  it 
thereto.  Newton's  LETTER  TO  A  PERSON  OF  DISTINCTION  WHO 

HAD  DESIRED   HIS  OPINION  OF   THE    LEARNED    BlSHO^    LLOYD'S 

HYPOTHESIS  CONCERNING  THE  FORM  OF  THE  MOST  ANCIENT 
^EAR,  closes  this  enumeration  of  his  Chronological  Writings. 

A  ihird  edition  of  the  PRINCIPIA  appeared  in  1726,  with  many 
changes  and  additions.  About  four  years  were  consumed  in  its 
preparation  and  publication,  which  were  under  the  superintend- 
ance  of  Dr.  Henry  Pemberton,  an  accomplished  mathematician, 
and  the  author  of  "A  VIEW  OF  SIR  ISAAC  NEWTON'S  PHILO 
SOPHY."  1728.  This  gentleman  enjoyed  numerous  opportunities 
of  conversing  with  the  aged  and  illustrious  author.  "  I  found," 
says  Pemberton,  "  he  had  read  fewer  of  the  modern  mathemati 
cians  than  one  could  have  expected;  but  his  own  prodigious 
invention  readily  supplied  him  with  what  he  might  have  an  occa 
sion  for  in  the  pursuit  of  any  subject  he  undertook.  I  have  often 


LIFE    OF    SIR    ISAAC    NEWTON.  57 

heard  him  censure  the  handling  geometrical  subjects  ly  algebraic 
calculations ;  and  his  book  of  Algebra  he  called  by  the  name  of 
Universal  Arithmetic,  in  opposition  to  the  injudicious  title  of 
Geometry,  which  Descartes  had  given  to  the  treatise,  wherein  he 
shows  how  the  geometer  may  assist  his  invention  by  such  kind 
of  computations.  He  thought  Huygens  the  most  elegant  of  any 
mathematical  writer  of  modern  times,  and  the  most  just  imitator 
of  the  ancients.  Of  their  taste  and  form  of  demonstration,  Sir 
Isaac  always  professed  himself  a  great  admirer.  I  have  heard 
him  even  censure  himself  for  not  following  them  yet  more  closely 
than  he  did  ;  and  speak  with  regret  of  his  mistake  at  the  begin 
ning  of  his  mathematical  studies,  in  applying  himself  to  the  works 
of  Descartes  and  other  algebraic  writers,  before  he  had  considered 
the  elements  of  Euclid  with  that  attention  which  so  excellent  a 
writer  deserves." 

"  Though  his  memory  was  much  decayed,"  continues  Dr.  Pem- 
berton,  "he  perfectly  understood  his  own  writings."  And  even 
this  failure  of  memory,  we  would  suggest,  might  have  been  more 
apparent  than  real,  or,  in  medical  terms,  more  the  result  of  func 
tional  weakness  than  organic  decay.  Newton  seems  never  to 
have  confided  largely  to  his  memory :  and  as  this  faculty  mani 
fests  the  most  susceptibility  to  cultivation  ;  so,  in  the  neglect  of 
due  exercise,  it  more  readily  and  plainly  shows  a  diminution  of 
its  powers. 

Equanimity  and  temperance  had,  indeed,  preserved  Newton 
singularly  free  from  all  mental  and  bodily  ailment.  His  hair  was, 
to  the  last,  quite  thick,  though  as  white  as  silver.  He  never 
made  use  of  spectacles,  and  lost  but  one  tooth  to  the  day  of  his 
death.  He  was  of  middle  stature,  well-knit,  and,  in  the  latter 
part  of  his  life,  somewhat  inclined  to  be  corpulent.  Mr.  Conduit 
says,  "  he  had  a  very  lively  and  piercing  eye,  a  comely  and  gra 
cious  aspect,  with  a  fine  head  of  hair,  white  as  silver,  without  any 
baldness,  and  when  his  peruke  was  off  was  a  venerable  sight." 
According  to  Bishop  Atterbury,  "in  the  whole  air  of  his  face  and 
make  there  was  nothing  of  that  penetrating  sagacity  which 
appears  in  his  compositions.  He  had  something  rather  languid 
in  his  look  and  manner  which  did  not  raise  any  great  expectation 


58  LIFE    OF    SIR    ISAAC    NEWTON. 

in  those  who  did  not  know  him."  Hearne  remarks,  "  Sir  Isaac 
was  a  man  of  no  very  promising  aspect.  He  was  a  short,  well- 
set  man.  He  was  full  of  thought,  and  spoke  very  little  in  com 
pany,  so  that  his  conversation  was  not  agreeable.  When  he  rode 
in  his  coach,  one  arm  would  be  out  of  his  coach  on  one  side  and 
the  other  on  the  other."  These  different  accounts  we  deem 
easily  reconcilable.  In  the  rooms  of  the  Royal  Society,  in  the 
street,  or  in  mixed  assemblages,  Newton's  demeanour — always 
courteous,  unassuming  and  kindly — still  had  in  it  the  overawings 
of  a  profound  repose  and  reticency,  out  of  which  the  communica 
tive  spirit,  and  the  "lively  and  piercing  eye"  would  only  gleam 
in  the  quiet  and  unrestrained  freedom  of  his  own  fire-side. 

"  But  this  I  immediately  discovered  in  him,"  adds  Pemberton, 
still  further,  "which  at  once  both  surprised  and  charmed  me. 
Neither  his  extreme  great  age,  nor  his  universal  reputation  had 
rendered  him  stiff  in  opinion,  or  in  any  degree  elated.  Of  this  I 
had  occasion  to  have  almost  daily  experience.  The  remarks  I 
continually  sent  him  by  letters  on  his  Principia,  were  received 
with  the  utmost  goodness.  These  were  so  far  from  being  any 
ways  displeasing  to  him,  that,  on  the  contrary,  it  occasioned  him 
to  speak  many  kind  things  of  me  to  my  friends,  and  to  honour  me 
with  a  public  testimony  of  his  good  opinion."  A  modesty,  open 
ness,  and  generosity,  peculiar  to  the  noble  and  comprehensive 
spirit  of  Newton.  "  Full  of  wisdom  and  perfect  in  beauty,"  yet 
not  lifted  up  by  pride  nor  corrupted  by  ambition.  None,  how 
ever,  knew  so  well  as  himself  the  stupendousness  of  his  discoveries 
in  comparison  with  all  that  had  been  previously  achieved ;  and 
none  realized  so  thoroughly  as  himself  the  littleness  thereof  in 
comparison  with  the  vast  region  still  unexplored.  A  short  time 
before  his  death  he  uttered  this  memorable  sentiment: — "  I  do  not 
know  what  I  may  appear  to  the  world ;  but  to  myself  I  seem  to 
have  been  only  like  a  boy  playing  on  the  sea-shore,  and  diverting 
myself  in  now  and  then  finding  a  smoother  pebble  or  a  prettier 
shell  than  ordinary,  while  the  great  ocean  of  truth  lay  all  undis 
covered  before  me."  How  few  ever  reach  the  shore  even,  much 
less  find  "a  smoother  pebble  or  a  prettier  shell!" 

Newton  had  now  resided  about  two  years  at  Kensington ;  and 


LIFE    OF    SIR    ISAAC    NEWTON.  59 

the  air  which  he  enjoyed  there,  and  the  state  of  absolute  rest, 
proved  of  great  benefit  to  him.  Nevertheless  he  would  occasion 
ally  go  to  town.  And  on  Tuesday,  the  28th  of  February,  1727, 
he  proceeded  to  London,  for  the  purpose  of  presiding  at  a  meeting 
of  the  Royal  Society.  At  this  time  his  health  was  considered, 
by  Mr.  Conduit,  better  than  it  had  been  for  many  years.  But 
the  unusual  fatigue  he  was  obliged  to  suffer,  in  attending  the 
meeting,  and  in  paying  and  receiving  visits,  speedily  produced  a 
violent  return  of  the  affection  in  the  bladder.  He  returned  to 
Kensington  on  Saturday,  the  4th  of  March.  Dr.  Mead  and  Dr. 
Cheselden  attended  him ;  they  pronounced  his  disease  to  be  the 
stone,  and  held  out  no  hopes  of  recovery.  On  Wednesday,  the 
15th  of  March,  he  seemed  a  little  better;  and  slight,  though 
groundless,  encouragement  was  felt  that  he  might  survive  the 
attack.  From  the  very  first  of  it,  his  sufferings  had  been  intense. 
Paroxysm  followed  paroxysm,  in  quick  succession  :  large  drops 
)f  sweat  rolled  down  his  face  ;  but  not  a  groan,  not  a  complaint, 
not  the  least  mark  of  peevishness  or  impatience  escaped  him  : 
and  during  the  short  intervals  of  relief,  he  even  smiled  and  con 
versed  with  his  usual  composure  and  cheerfulness.  The  flesh 
quivered,  but  the  heart  quaked  not ;  the  impenetrable  gloom  was 
settling  down :  the  Destroyer  near ;  the  portals  of  the  tomb 
opening,  still,  arnid  this  utter  wreck  and  dissolution  of  the  mortal, 
the  immortal  remained  serene,  unconquerable :  the  radiant  light 
broke  through  the  gathering  darkness ;  and  Death  yielded  up  its 
sting,  and  the  grave  its  victory.  On  Saturday  morning,  18th, 
he  read  the  newspapers,  and  carried  on  a  pretty  long  conversation 
with  Dr.  Mead.  His  senses  and  faculties  were  then  strong  and 
vigorous  ;  but  at  six  o'clock,  the  same  evening,  he  became  insen 
sible  ;  and  in  this  state  he  continued  during  the  whole  of  Sunday, 
and  till  Monday,  the  20th,  when  he  expired,  between  one  and 
two  o'clock  in  the  morning,  in  the  eighty-fifth  year  of  his  age. 

And  these  were  the  last  days  of  Isaac  Newton.  Thus  closed 
the  career  of  one  of  earth's  greatest  and  best  men.  His  mission 
was  fulfilled.  Unto  the  Giver,  in  many-fold  addition,  the  talents 
were  returned.  While  it  was  yet  day  he  had  worked ;  and  for 
the  night  that  quickly  cometh  he  was  not  unprepared.  Full  of 


60  LIFE    OF    SIR    ISAAC    NEWTON. 

years,  ind  full  of  honours,  the  heaven-sent  was  recalled ;  and,  in 
the  confidence  of  a  "  certain  hope,"  peacefully  he  passed  awa}' 
into  the  silent  depths  of  Eternity. 

His  body  was  placed  in  Westminster  Abbey,  with  the  state 
and  ceremonial  that  usually  attended  the  interment  of  the  most 
distinguished.  In  1731,  his  relatives,  the  inheritors  of  his  personal 
estate,  erected  a  monument  to  his  memory  in  the  most  conspicu 
ous  part  of  the  Abbey,  which  had  often  been  refused  by  the  dean 
and  chapter  to  the  greatest  of  England's  nobility.  During  the 
same  year  a  medal  was  struck  at  the  Tower  in  his  honour  ;  arid, 
in  1755,  a  full-length  statue  of  him,  in  white  marble,  admirably 
executed,  by  Roubiliac,  at  the  expense  of  Dr.  Robert  Smith,  was 
erected  in  the  ante-chamber  of  Trinity  College,  Cambridge. 
There  is  a  painting  executed  in  the  glass  of  one  of  the  windows 
of  the  same  college,  made  pursuant  to  the  will  of  Dr.  Smith,  who 
left  five  hundred  pounds  for  that  purpose. 

Newton  left  a  personal  estate  of  about  thirty-two  thousand 
pounds.  It  was  divided  among  his  four  nephews  and  four  nieces 
of  the  half  blood,  the  grand-children  of  his  mother,  by  the  Reve 
rend  Mr.  Smith.  The  family  estates  of  Woolsthorpe  arid  Sustern 
fell  to  John  Newton,  the  heir-at-law,  whose  great  grand-father 
was  Sir  Isaac's  uncle.  Before  his  death  he  made  an  equitable 
distribution  of  his  two  other  estates :  the  one  in  Berkshire  to  the 
sons  and  daughter  of  a  brother  of  Mrs.  Conduit ;  and  the  other, 
at  Kensington,  to  Catharine,  the  only  daughter  of  Mr.  Conduit, 
and  who  afterward  became  Viscountess  Lymington.  Mr.  Con 
duit  succeeded  to  the  offices  of  the  Mint,  the  duties  of  which  he 
had  discharged  during  the  last  two  years  of  Sir  Isaac's  life. 

Our  author's  works  are  found  in  the  collection  of  Castilion, 
Berlin,  1744,  4to.  8  torn.;  in  Bishop  Horsley's  Edition,  London, 
1779,  4to.  5  vol.;  in  the  Biographia  Brittannica,  &c.  Newton 
also  published  Bern.  Varcnii  Geographia,  &c.,  1681,  8vo. 
There  are,  however,  numerous  manuscripts,  letters,  and  other 
papers,  which  have  never  been  given  to  the  world:  these  are 
preserved,  in  various  collections,  namely,  in  the  library  of  Trinity 
College,  Cambridge ;  in  the  library  of  Corpus  Christi  College, 
Oxford ;  in  the  library  of  Lord  Macclesfield :  and,  lastly  arid 


LIFE    OF    SIR    ISAAC    NEWTON.  61 

chiefly,  in  the  possession  of  the  family  of  the  Earl  of  Portsmouth, 
through  the  Viscountess  Lymington. 

Everything  appertaining  to  Newton  has  been  kept  and  che 
rished  with  peculiar  veneration.  Different  memorials  of  him  are 
preserved  in  Trinity  College,  Cambridge  ;  in  the  rooms  of  the 
Royal  Society,  of  London  :  and  in  the  Museum  of  the  Royal 
Society  of  Edinburgh. 

The  manor-house,  at  Woolsthorpe,  was  visited  by  Dr.  Stuke 
ley,  in  October,  1721,  who,  in  a  letter  to  Dr.  Mead,  written  in 
1727,  gave  the  following  description  of  it: — " 'Tis  built  of  stone, 
as  is  the  way  of  the  country  hereabouts,  and  a  reasonably  good 
one.  They  led  me  up  stairs  and  showed  me  Sir  Isaac's  stud}-, 
where  I  supposed  he  studied,  when  in  the  country,  in  his  younger 
days,  or  perhaps  when  he  visited  his  mother  from  the  University. 
I  observed  the  shelves  were  of  his  own  making,  being  pieces  of 
deal  boxes,  which  probably  he  sent  his  books  and  clothes  down 
in  on  those  occasions.  There  were,  some  years  ago,  two  or  threr 
hundred  books  in  it  of  his  father-in-law,  Mr.  Smith,  which  Sir 
Isaac  gave  to  Dr.  Newton,  of  our  town."  The  celebrated  apple- 
tree,  the  fall  of  one  of  the  apples  of  which  is  said  to  have  turned 
the  attention  of  Newton  to  the  subject  of  gravity,  was  destroyed 
by  the  wind  about  twenty  years  ago ;  but  it  has  been  preserved 
in  the  form  of  a  chair.  The  house  itself  has  been  protected  with 
religious  care.  It  was  repaired  in  1798,  and  a  tablet  of  white 
marble  put  up  in  the  room  where  our  author  was  born,  with  the 
follow, ng  inscription  : — 

"  Sir  Isaac  Newton,  son  of  John  Newton,  Lord  of  the  Manor 
of  Woolsthorpe,  was  born  in  this  room,  on  the  25th  of  December, 
1642." 

Nature  and  Nature's  Laws  wei-e  hid  in  night, 
God  said,  "  Let  NEWTON  be,"  and  all  was  light. 


THE    PEINCIPIA. 


THE     AUTHOR'S    PREFACE 

SINCE  the  ancients  (as  we  are  told  by  Pappus),  made  great  account  oi 
the  science  of  mechanics  in  the  investigation  of  natural  things :  and  the 
moderns,  laying  aside  substantial  forms  and  occult  qualities,  have  endeav 
oured  to  subject  the  phenomena  of  nature  to  the  laws  of  mathematics,  I 
have  in  this  treatise  cultivated  mathematics  so  far  as  it  regards  philosophy. 
The  ancients  considered  mechanics  in  a  twofold  respect ;  as  rational,  which 
proceeds  accurately  by  demonstration  ;  and  practical.  To  practical  me 
chanics  all  the  manual  arts  belong,  from  which  mechanics  took  its  name. 
Rut  as  artificers  do  not  work  with  perfect  accuracy,  it  comes  to  pass  that 
mechanics  is  so  distinguished  from  geometry,  that  what  is  perfectly  accu 
rate  is  called  geometrical ,  what  is  less  so,  is  called  mechanical.  But  the 
errors  are  not  in  the  art,  but  in  the  artificers.  He  that  works  with  less 
accuracy  is  an  imperfect  mechanic ;  and  if  any  could  work  with  perfect 
accuracy,  he  would  be  the  most  perfect  mechanic  of  all ;  for  the  description 
if  right  lines  and  circles,  upon  which  geometry  is  founded,  belongs  to  me 
chanics.  Geometry  does  not  teach  us  to  draw  these  lines,  but  requires 
them  to  be  drawn  ;  for  it  requires  that  the  learner  should  f.rst  be  taught 
to  describe  these  accurately,  before  he  enters  upon  geometry  ;  then  it  shows 
how  by  these  operations  problems  may  be  solved.  To  describe  right  lines 
and  circles  are  problems,  but  not  geometrical  problems.  The  solution  of 
these  problems  is  required  from  mechanics ;  and  by  geometry  the  use  of 
them,  when  so  solved,  is  shown  ;  and  it  is  the  glory  of  geometry  that  from 
those  few  principles,  brought  from  without,  it  is  able  to  produce  so  many 
things.  Therefore  geometry  is  founded  in  mechanical  practice,  and  is 
nothing  but  that  part  of  universal  mechanics  which  accurately  proposes 
and  demonstrates  the  art  of  measuring.  But  since  the  manual  arts  are 
chiefly  conversant  in  the  moving  of  bodies,  it  comes  to  pass  that  geometry 
is  commonly  referred  to  their  magnitudes,  and  mechanics  to  their  motion. 
In  this  sense  rational  mechanics  will  be  the  science  of  motions  resulting 
from  any  forces  whatsoever,  and  of  the  forces  required  to  produce  any  mo 
tions,  accurately  proposed  and  demonstrated.  This  part  of  mechanics  was 


i:;vm  THE  AUTHOR  &  PREFACE. 

cultivated  by  the  ancients  in  the  five  powers  which  relate  to  manual  arts, 
who  considered  gravity  (it  not  being  a  manual  power),  ho  Otherwise  than 
as  it  moved  weights  by  those  powers.  Our  design  not  respecting  arts,  but 
philosophy,  and  our  subject  not  manual  but  natural  powers,  we  consider 
chiefly  those  things  which  relate  to  gravity,  levity,  elastic  force,  the  resist 
ance  of  fluids,  and  the  like  forces,  whether  attractive  or  impulsive ;  and 
therefore  we  offer  this  work  as  the  mathematical  principles  :f  philosophy ;  for 
all  the  difficulty  of  philosophy  seems  to  consist  in  this — from  the  phenom 
ena  of  motions  to  investigate  the  forces  of  nature,  and  then  from  these 
forces  to  demonstrate  the  other  phenomena ;  and  to  this  end  the  general 
propositions  in  the  first  and  second  book  are  directed.  In  the  third  book 
we  give  an  example  of  this  in  the  explication  of  the  System  of  the  World  : 
for  by  the  propositions  mathematically  demonstrated  in  the  former  books, 
we  in  the  third  derive  from  the  celestial  phenomena  the  forces  of  gravity 
with  which  bodies  tend  to  the  sun  and  the  several  planets.  Then  from  these 
forces,  by  other  propositions  which  are  also  mathematical,  we  deduce  the  mo 
tions  of  the  planets,  the  comets,  the  moon,  and  the  sea.  I  wish  we  could  do- 
rive  the  rest  of  the  phenomena  of  nature  by  the  same  kind  of  reasoning  from 
mechanical  principles;  for  I  am  induced  by  many  reasons  to  suspect  that 
they  may  all  depend  upon  certain  forces  by  which  the  particles  of  bodies. 
by  some  causes  hitherto  unknown,  are  either  mutually  impelled  towards 
each  other,  and  cohere  in  regular  figures,  or  are  repelled  and  recede  from 
each  other;  which  forces  being  unknown,  philosophers  have  hitherto  at 
tempted  the  search  of  nature  in  vain ;  but  I  hope  the  principles  here  laid 
down  will  afford  some  light  either  to  this  or  some  truer  method  of  philosophy. 
In  the  publication  of  this  work  the  most  acute  and  universally  learned 
Mr.  Edmund  H alley  not  only  assisted  me  with  his  pains  in  correcting  the 
press  and  taking  care  of  the  schemes,  but  it  was  to  his  solicitations  that  its 
becoming  public  is  owing ;  for  when  he  had  obtained  of  me  my  demonstra 
tions  of  the  figure  of  the  celestial  orbits,  he  continually  pressed  me  to  com 
municate  the  same  to  the  Royal  Societ //,  who  afterwards,  by  their  kind  en 
couragement  and  entreaties,  engaged  me  to  think  of  publishing  them.  But 
after  I  had  begun  to  consider  the  inequalities  of  the  lunar  motions,  and 
had  entered  upon  some  other  things  relating  to  the  laws  and  measures  oi 
gravity,  and  other  forces  :  and  the  figures  that  would  be  described  by  bodies 
attracted  according  to  given  laws  ;  and  the  motion  of  several  bodies  moving 
among  themselves;  the  motion  of  bodies  in  resisting  mediums;  the  forces, 
densities,  and  motions,  of  rn(  Hums ;  the  orbits  of  the  comets,  and  such  like ; 


Ixix 

deferred  that  publication  till  I  had  made  a  searcli  into  those  matters,  and 
could  put  forth  the  whole  together.  What  relates  to  the  lunar  motions  (be 
ing  imperfect),  I  have  put  all  together  in  the  corollaries  of  Prop.  66,  to 
avoid  being  obliged  to  propose  and  distinctly  demonstrate  the  several  things 
there  contained  in  a  method  more  prolix  than  the  subject  deserved,  and  in 
terrupt  the  series  of  the  several  propositions.  Some  things,  found  out  after 
the  rest,  I  chose  to  insert  in  places  less  suitable,  rather  than  change  the 
number  of  the  propositions  and  the  citations.  I  heartily  beg  that  what  1 
have  here  done  may  be  read  with  candour;  and  that  the  defects  in  a 
subject  so  difficult  be  not  so  much  reprehended  as  kindly  supplied,  and  in 
vestigated  by  new  endeavours  of  mv  readers. 

ISAAC  NEWTON. 

Cambridge,  Trinity  Coupge    May  8,  l«iHB. 

In  the  second  edition  the  second  section  of  the  first  book  was  enlarged. 
In  the  seventh  section  of  the  second  book  the  theory  of  the  resistances  of  fluids 
was  more  accurately  investigated,  and  confirmed  by  new  experiments.  In 
the  third  book  the  moon's  theory  and  the  profession  of  the  equinoxes  were 
more  fully  deduced  from  their  principles ;  and  the  theory  of  the  comets 
was  confirmed  by  more  examples  of  the  calculati  >n  of  their  orbits,  done 
also  with  greater  accuracy. 

In  this  third  edition  the  resistance  of  mediums  is  somewhat  more  largely 
handled  than  before;  and  new  experiments  of  the  resistance  of  heavy 
bodies  falling  in  air  are  added.  In  the  third  book,  the  argument  to  prove 
that  the  moon  is  retained  in  its  orbit  by  the  force  of  gravity  is  enlarged 
on ;  and  there  are  added  new  observations  of  Mr.  Pound's  of  the  proportion 
of  the  diameters  of  Ju.piter  to  each  other :  there  are,  besides,  added  Mr. 
Kirk's  observations  of  the  comet  in  16SO ;  the  orbit  of  that  comet  com 
puted  in  an  ellipsis  by  Dr.  Halley ;  and  the  ortit  of  the  comet  in 
computed  by  Mr.  Bradley, 


OOK   I. 


THE 

MATHEMATICAL  PRINCIPLES 


OF 


NATURAL    PHILOSOPHY 


DEFINITIONS. 

DEFINITION  I. 
77w?  quantity  of  matter  is  the  measure  of  the  same,  arising  from  its 

density  and  hulk  conjutictly. 

THUS  air  of  a  double  density,  in  a  double  space,  is  quadruple  in  quan- 
ti  ty ;  in  a  triple  space,  sextuple  in  quantity.  The  same  thing  is  to  be  un 
derstood  of  snow,  and  fine  dust  or  powders,  that  are  condensed  by  compres 
sion  or  liquefaction  •  and  of  all  bodies  that  are  by  any  causes  whatever 
differently  condensed.  I  have  no  regard  in  this  place  to  a  medium,  if  any 
such  there  is,  that  freely  pervades  the  interstices  between  the  parts  oi 
bodies.  It  is  this  quantity  that  I  mean  hereafter  everywhere  under  the 
name  of  body  or  mass.  And  the  same  is  known  by  the  weight  of  each 
body ;  for  it  is  proportional  to  the  weight,  as  I  have  found  by  experiments 
on  pendulums,  very  accurately  made,  which  shall  be  shewn  hereafter. 

DEFINITION  II. 

The  quantity  of  motion  is  the  measure  nf  tlie  same,  arising  from  the 

velocity  and  quantity  of  matter  corjunctly. 

The  motion  of  the  whole  i<!  the  sum  of  the  motions  of  all  the  parts ;  and 
therefore  in  a  body  double  in  quantity,  with  equal  velocity,  the  motion  is 
iouble ;  with  twice  the  velocity,  it  is  quadruple, 

DEFINITION  III. 

The  vis  insita,  or  innate  force  of  matter,  is  a  power  of  resisting,  hy 
which  every  body,  as  much  as  in  it  lies,  endeavours  to  persevere  in  its 
present  stale,  whether  it  be  of  rest,  or  of  moving  uniformly  forward 
in  a  right  line. 
This  force  is  ever  proportional  to  the  body  whose  force  it  is ;  and  differs 

nothing  from  the  inactivity  of  the  mass,  but  in  our  manner  of  conceiving 


T4  THE   MATHEMATICAL    PRINCIPLES 

it.  A  body,  from  the  inactivity  of  matter,  is  not  without  difficulty  put  out 
of  its  state  of  rest  or  motion.  Upon  which  account,  this  vis  insita,  may, 
by  a  most  significant  name,  be  called  vis  inertia,  or  force  of  inactivity. 
Hut  a  body  exerts  this  force  only,  when  another  force,  impressed  upon  it, 
endeavours  to  change  its  condition ;  and  the  exercise  of  this  force  may  bo 
considered  both  as  resistance  and  impulse ;  it  is  resistance,  in  so  far  as  the 
body,  for  maintaining  its  present  state,  withstands  the  force  impressed;  it 
is  impulse,  in  so  far  as  the  body,  by  not  easily  giving  way  to  the  impressed 
force  of  another,  endeavours  to  change  the  state  of  that  other.  Resistance 
is  usually  ascribed  to  bodies  at  rest,  and  impulse  to  those  in  motion; 
but  motion  and  rest,  as  commonly  conceived,  are  only  relatively  distin 
guished  ;  nor  are  those  bodies  always  truly  at  rest,  which  commonly  are 
taken  to  be  so. 

DKFLMTIOX  IV. 

Ait  impressed  force  is  an  action  exerted  upon  a  body,  in  order  to  change 
its  state,  either  of  rest,  or  of  moving  uniformly  forward  in  a  right 
line. 

This  force  consists  in  the  action  only;  and  remains  no  longer  in  the 
body,  when  the  action  is  over.  For  a  body  maintains  every  new  state  it 
acquires,  by  its  vis  inertice  only.  Impressed  forces  are  of  differe.it  origins  • 
as  from  percussion,  from  pressure,  from  centripetal  force. 

DEFINITION  V. 

A  centripetal  force  is  that  by  which  bodies  are  drawn  or  impelled,  or  any 

way  tend,  towards  a  point  as  to  a  centre. 

Of  this  sort  is  gravity,  by  which  bodies  tend  to  the  centre  of  the  earth 
magnetism,  by  which  iron  tends  to  the  loadstone ;  and  that  force,  what 
ever  it  is,  by  which  the  planets  are  perpetually  drawn  aside  from  the  rec 
tilinear  motions,  which  otherwise  they  would  pursue,  and  made  to  revolve 
in  curvilinear  orbits.  A  stone,  whirled  about  in  a  sling,  endeavours  to  re 
cede  from  the  hand  that  turns  it ;  and  by  that  endeavour,  distends  the 
sling,  and  that  with  so  much  the  greater  force,  as  it  is  revolved  with  the 
greater  velocity,  and  as  soon  as  ever  it  is  let  go,  flies  away.  That  force 
which  opposes  itself  to  this  endeavour,  and  by  which  the  sling  perpetually 
draws  back  the  stone  towards  the  hand,  and  retains  it  in  its  orbit,  because 
it  is  directed  to  the  hand  as  the  centre  of  the  orbit,  I  call  the  centripetal 
force.  And  the  same  thing  is  to  be  understood  of  all  bodies,  revolved  in 
any  orbits.  They  all  endeavour  to  recede  from  the  centres  of  their  orbits ; 
and  wore  it  not  for  the  opposition  of  a  contrary  force  which  restrains  them 
to,  and  detains  them  in  their  orbits,  which  I  therefore  call  centripetal,  would 
tiy  off  in  right  lines,  with  an  uniform  motion.  A  projectile,  if  it  was  not 
for  the  force  of  gravity,  would  not  deviate  towards  the  earth,  tut  would 


OF    NATUJIAL    PHILOSOPHY.  7fl 

go  off  from  it  in  a  right  line,  and  that  with  an  uniform  motion,, if  the  re 
sistance  of  the  air  was  taken  away.  It  is  by  its  gravity  that  it  is  drawn 
aside  perpetually  from  its  rectilinear  course,  and  made  to  deviate  towards 
the  earth,  more  or  less,  according  to  the  force  of  its  gravity,  and  the  velo 
city  of  its  motion.  The  less  its  gravity  is,  for  the  quantity  of  its  matter, 
or  the  greater  the  velocity  with  which  it  is  projected,  the  less  will  it  devi 
ate  from  a  rectilinear  course,  and  the  farther  it  will  go.  If  a  leaden  balJ, 
projected  from  the  top  of  a  mountain  by  the  force  of  gunpowder  with  a 
given  velocity,  and  in  a  direction  parallel  to  the  horizon,  is  carried  in  a 
curve  line  to  the  distance  of  two  miles  before  it  falls  to  the  ground  ;  the 
same,  if  the  resistance  of  the  air  were  taken  away,  with  a  double  or  decuple 
velocity,  would  fly  twice  or  ten  times  as  far.  And  by  increasing  the  velo 
city,  we  may  at  pleasure  increase  the  distance  to  which  it  might  be  pro 
jected,  and  diminish  the  curvature  of  the  line,  which  it  might  describe,  till 
at  last  it  should  fall  at  the  distance  of  10,  30,  or  90  degrees,  or  even  might 
go  quite  round  the  whole  earth  before  it  falls ;  or  lastly,  so  that  it  might 
never  fall  to  the  earth,  but  go  forward  into  the  celestial  spaces,  and  pro 
ceed  in  its  motion  in  iiifiuitum.  And  after  the  same  manner  that  a  pro 
jectile,  by  the  force  of  gravity,  may  be  made  to  revolve  in  an  orbit,  and  go 
round  the  whole  earth,  the  moon  also,  either  by  the  force  of  gravity,  if  it 
is  endued  with  gravity,  or  by  any  other  force,  that  impels  it  towards  the 
earth,  may  be  perpetually  drawn  aside  towards  the  earth,  out  of  the  r&ti- 
linear  way,  which  by  its  innate  force  it  would  pursue;  and  would  be  made 
to  revolve  in  the  orbit  which  it  now  describes ;  nor  could  the  moon  with 
out  some  such  force,  be  retained  in  its  orbit.  If  this  force  was  too  small, 
it  would  not  sufficiently  turn  the  moon  out  of  a  rectilinear  course :  if  it 
was  too  great,  it  would  turn  it  too  much,  arid  draw  down  the  moon  from 
its  orbit  towards  the  earth.  It  is  necessary,  that  the  force  be  of  a  just 
quantity,  and  it  belongs  to  the  mathematicians  to  find  the  force,  that  may 
serve  exactly  to  retain  a  body  in  a  given  orbit,  with  a  given  velocity ;  and 
vice  versa,  to  determine  the  curvilinear  way,  into  which  a  body  projected 
from  a  given  place,  with  a  given  velocity,  may  be  made  to  deviate  from 
its  natural  rectilinear  way,  by  means  of  a  given  force. 

The  quantity  of  any  centripetal  force  may  be  considered  as  of  three 
kinds;  aboolu'e,  accelerative,  and  motive. 

DEFINITION  VI. 

The  absolute  quantity  of  a  centripetal  force  is  the  measure  f>f  the  same 
proportional  to  the  efficacy  of  the  cause  that  propagates  it  from  the  cen 
tre,  through  the  spaces  round  about. 
Thus  the  magnetic  force  is  greater  in  one  load-stone  and  less  in  another 

according  to  their  sizes  and  strength  of  intensity. 


76  THE    MATHEMATICAL    PRINCIPLES 

DEFINITION  VII. 

The  accelerative  quantity  of  a  centripetal  force  is  the  measure,  of  tht 
same,  proportional  to  the  velocity  which  it  generates  in  a  given  time. 

Thus  the  force  of  the  same  load-stone  is  greater  at  a  less  distance,  and 
less  at  a  greater :  also  the  force  of  gravity  is  greater  in  valleys,  less  on 
tops  of  exceeding  high  mountains ;  and  yet  less  (as  shall  hereafter  be  shown), 
at  greater  distances  from  the  body  of  the  earth ;  but  at  equal  distan 
ces,  it  is  the  same  everywhere ;  because  (taking  away,  or  allowing  for,  the 
resistance  of  the  air),  it  equally  accelerates  all  falling  bodies,  whether  heavy 
or  light,  great  or  small. 

DEFINITION  VIII. 

TJie  motive  quantity  of  a  centripetal  force,  is  the  measure  of  the  samt\ 

proportional  to  the  motion  which  it  generates  in  a  given  twip. 

Thus  the  weight  is  greater  in  a  greater  body,  less  in  a  less  body ;  and. 
in  the  same  body,  it  is  greater  near  to  the  earth,  and  less  at  remoter  dis 
tances.  This  sort  of  quantity  is  the  centripetency,  or  propension  of  the 
whole  body  towards  the  centre,  or,  as  I  may  say,  its  weight ;  and  it  is  al 
ways  known  by  the  quantity  of  an  equal  and  contrary  force  just  sufficient 
to  Ifinder  the  descent  of  the  body. 

These  quantities  of  forces,  we  may,  for  brevity's  sake,  call  by  the  names 
of  motive,  accelerative,  and  absolute  forces ;  and,  for  distinction's  sake,  con 
sider  them,  with  respect  to  the  bodies  that  tend  to  the  centre ;  to  the  places 
of  those  bodies ;  and  to  the  centre  of  force  towards  which  they  tend  ;  that 
is  to  say,  I  refer  the  motive  force  to  the  body  as  an  endeavour  and  propen 
sity  of  the  whole  towards  a  centre,  arising  from  the  propensities  of  the 
several  parts  taken  together ;  the  accelerative  force  to  the  place  of  the 
body,  as  a  certain  power  or  energy  diffused  from  the  centre  to  all  places 
around  to  move  the  bodies  that  are  in  them :  and  the  absolute  force  to 
the  centre,  as  endued  with  some  cause,  without  which  those  motive  forces 
would  not  be  propagated  through  the  spaces  round  about ;  whether  that 
cause  be  some  central  body  (siuh  as  is  the  load-stone,  in  the  centre  of  the 
magnetic  force,  or  the  earth  in  the  centre  of  the  gravitating  force),  or 
anything  else  that  does  not  yet  appear.  For  I  here  design  only  to  give  a 
mathematical  notion  of  those  forces,  without  considering  their  physical 
causes  and  seats. 

Wherefore  the  accelerative  force  will  stand  in  the  same  relation  to  the 
motive,  as  celerity  does  to  motion.  For  the  quantity  of  motion  arises  from 
the  celerity  drawn  into  the  quantity  of  matter :  and  the  motive  force  arises 
from  the  accelerative  force  drawn  into  the  same  quantity  of  matter.  For 
the  sum  of  the  actions  of  the  accelerative  force,  upon  the  several  ;  articles 
of  the  body,  is  the  motive  force  of  the  whole.  Hence  it  is,  that  near  the 


OF    NATURAL    PHILOSOPHY.  77 

surface  of  the  earth,  where  the  accelerative  gravity,  or  force  productive  of 
gravity,  in  all  bodies  is  the  same,  the  motive  gravity  or  the  weight  is  as 
the  body :  but  if  we  should  ascend  to  higher  regions,  where  the  accelerative 
gravity  is  less,  the  weight  would  be  equally  diminished,  and  would  always 
be  as  the  product  of  the  body,  by  the  accelerative  gravity.  So  in  those  re 
gions,  where  the  accelerative  gravity  is  diminished  into  one  half,  the  weight 
of  a  body  two  or  three  times  less,  will  be  four  or  six  times  less. 

I  likewise  call  attractions  and  impulses,  in  the  same  sense,  accelerative, 
and  motive ;  and  use  the  words  attraction,  impulse  or  propensity  of  any 
sort  towards  a  centre,  promiscuously,  and  indifferently,  one  for  another ; 
considering  those  forces  not  physically,  but  mathematically  :  wherefore,  the 
reader  is  not  to  imagine,  that  by  those  words,  I  anywhere  take  upon  me  to 
define  the  kind,  or  the  manner  of  any  action,  the  causes  or  the  physical 
reason  thereof,  or  that  I  attribute  forces,  in  a  true  and  physical  sense,  to 
certain  centres  (which  are  only  mathematical  points) ;  when  at  any  time  I 
happen  to  speak  of  centres  as  attracting,  or  as  endued  with  attractive 
powers. 

SCHOLIUM. 

Hitherto  I  have  laid  down  the  definitions  of  such  words  as  are  less 
known,  and  explained  the  sense  in  which  I  would  have  them  to  be  under 
stood  in  the  following  discourse.  I  do  not  define  time,  space,  place  and 
motion,  as  being  well  known  to  all.  Only  I  must  observe,  that  the  vulgar 
conceive  those  quantities  under  no  other  notions  but  from  the  relation  they 
bear  to  sensible  objects.  And  thence  arise  certain  prejudices,  for  the  re 
moving  of  which,  it  will  be  convenient  to  distinguish  them  into  absolute 
and  relative,  true  and  apparent,  mathematical  and  common. 

I.  Absolute,  true,  and  mathematical  time,  of  itself,  and  from  its  own  na 
ture  flows  equably  without  regard  to  anything  external,  and  by  another 
name  is  called  duration  :  relative,  apparent,  and  common  time,  is  some  sen 
sible  and  external  (whether  accurate  or  unequable)  measure  of  duration  by 
the  means  of  motion,  which  is  commonly  used  instead  of  true  time ;  such 
as  an  hour,  a  day,  a  month,  a  year. 

II.  Absolute  space,  in  its  own  nature,  without  regard  to  anything  exter 
nal,  remains  always  similar  and  immovable.     Relative  space  is  some  mo 
vable  dimension  or  measure  of  the  absolute  spaces ;  which  our  senses  de 
termine  by  its  position  to  bodies ;  and  which  is  vulgarly  taken  for  immo 
vable  space ;  such  is  the  dimension  of  a  subterraneous,  an  aereal,  or  celestial 
space,  determined  by  its  position  in  respect  of  the  earth.     Absolute  and 
relative  space,  are  the  same  in  figure  and  magnitude ;  but  they  do  not  re 
main  always  numerically  the  same.     For  if  the  earth,  for  instance,  moves, 
a  space  of  our  air,  which  relatively  and  in  respect  of  the  earth  remains  al 
ways  the  same,  will  at  one  time  be  one  part  of  the  absolute  space  into  which 


TS  THE    MATHEMATICAL    PRINCIPLES 

the  air  passes  ;  at  another  time  it  will  be  another  part  of  the  same,  and  so. 
absolutely  understood,  it  will  be  perpetually  mutable. 

III.  Place  is  a  part  of  space  which  a  body  takes  up,  and  is  according  to 
the  space,  either  absolute  or  relative.   I  say,  a  part  of  space ;  not  the  situation, 
nor  the  external  surface  of  the  body.     For  the  places  of  equal  solids  are 
always  equal ;  but  their  superfices,  by  reason  of  their  dissimilar  figures,  are 
often  unequal.     Positions  properly  have  no  quantity,  nor  are  they  so  much 
the  places  themselves,  as  the  properties  of  places.     The  motion  of  the  whole 
is  the  same  thing  with  the  sum  of  the  motions  of  the  parts ;  that  is,  the 
translation  of  the  whole,  out  of  its  place,  is  the  same  thing  with  the  sum 
of  the  translations  of  the  parts  out  of  their  places ;  and  therefore  the  place 
of  the  whole  is  the  same  thing  with  the  sum  of  the  places  of  the  parts,  and 
for  that  reason,  it  is  internal,  and  in  the  whole  body. 

IV.  Absolute  motion  is  the  translation  of  a  body  from  one  absolute 
place  into  another ;  and  relative  motion,  the  translation  from  one  relative 
place  into  another.     Thus  in  a  ship  under  sail,  the  relative  place  of  a  body 
is  that  part  of  the  ship  which  the  body  possesses ;  or  that  part  of  its  cavity 
which  the  body  fills,  and  which  therefore  moves  together  with  the  ship : 
and  relative  rest  is  the  continuance  of  the  body  in  the  same  part  of  the 
ship,  or  of  its  cavity.     But  real,  absolute  rest,  is  the  continuance  of  the 
body  in  the  same  part  of  that  immovable  space,  in  which  the  ship  itself, 
its  cavity,  and  all  that  it  contains,  is  moved.     Wherefore,  if  the  earth  is 
really  at  rest,  the  body,  which  relatively  rests  in  the  ship,  will  really  and 
absolutely  move  with  the  same  velocity  which  the  ship  has  on  the  earth. 
But  if  the  earth  also  moves,  the  true  and  absolute  motion  of  the  body  will 
arise,  partly  from  the  true  motion  of  the  earth,  in  immovable  space ;  partly 
from  the  relative  motion  of  the  ship  on  the  earth ;  and  if  the  body  moves 
also  relatively  in  the  ship ;  its  true  motion  will  arise,  partly  from  the  true 
motion  of  the  earth,  in  immovable  space,  and  partly  from  the  relative  mo 
tions  as  well  of  the  ship  on  the  earth,  as  of  the  body  in  the  ship  ;  and  from 
these  relative  motions  will  arise  the  relative  motion  of  the  body  on  the 
earth.     As  if  that  part  of  the  earth,  where  the  ship  is,  was  truly  moved 
toward  the  east,  with  a  velocity  of  10010  parts;  while  the  ship  itself,  with 
a  fresh  gale,  and  full  sails,  is  carried  towards  the  west,  with  a  velocity  ex 
pressed  by  10  of  those  parts  ;  but  a  sailor  walks  in  the  ship  towards  the 
east,  with  1  part  of  the  said  velocity ;  then  the  sailor  will  be  moved  truly 
in  immovable  space  towards  the  east,  with  a  velocity  of  10001  parts,  and 
relatively  on  the  earth  towards  the  west,  with  a  velocity  of  9  of  those  parts. 

Absolute  time,  in  astronomy,  is  distinguished  from  relative,  by  the  equa 
tion  or  correction  of  the  vulgar  time.  For  the  natural  days  are  tr^y  un 
equal,  though  they  are  commonly  considered  as  equal,  and  used  for  a  meas 
ure  of  time ;  astronomers  correct  this  inequality  for  their  more  accurate 
deducing  of  the  celestial  motions.  It  may  be,  that  there  is  no  such  thing 
as  an  equable  motion,  whereby  time  may  H  accurately  measured.  All  mo 


OF    NATURAL    PHILOSOPHY.  79 

tions  may  be  accelerated  and  retarded;  but  the  true,  or  equable,  progress  of 
absolute  time  is  liable  to  no  change.  The  duration  or  perseverance  of  the 
existence  of  things  remains  the  same,  whether  the  motions  are  swift  or  slow, 
or  none  at  all :  and  therefore  it  ought  to  be  distinguished  from  what  are 
only  sensible  measures  thereof ;  and  out  of  which  we  collect  it,  by  means 
of  the  astronomical  equation.  The  necessity  of  which  equation,  for  deter 
mining  the  times  of  a  phamomenon,  is  evinced  as  well  from  the  experiments 
of  the  pendulum  clock,  as  by  eclipses  of  the  satellites  of  Jupiter. 

As  the  order  of  the  parts  of  time  is  immutable,  so  also  is  the  order  of 
the  parts  of  space.  Suppose  those  parts  to  be  moved  out  of  their  places,  and 
they  will  be  moved  (if  the  expression  may  be  allowed)  out  of  themselves. 
For  times  and  spaces  are,  as  it  were,  the  places  as  well  of  themselves  as  of 
all  other  things.  All  things  are  placed  in  time  as  to  order  of  succession ; 
and  in  space  as  _to  order  of  situation.  It  is  from  their  essence  or  nature 
that  they  are  places ;  and  that  the  primary  places  of  things  should  be 
moveable,  is  absurd.  These  are  therefore  the  absolute  places ;  and  trans 
lations  out  of  those  places,  are  the  only  absolute  motions. 

But  because  the  parts  of  space  cannot  be  seen,  or  distinguished  from  one 
another  by  our  senses,  therefore  in  their  stead  we  use  sensible  measures  of 
them.  For  from  the  positions  and  distances  of  things  from  any  body  con 
sidered  as  immovable,  we  define  all  places ;  and  then  with  respect  to  such 
places,  we  estimate  all  motions,  considering  bodies  as  transferred  from  some 
of  those  places  into  others.  And'so,  instead  of  absolute  places  and  motions, 
we  use  relative  ones;  and  that  without  any  inconvenience  in  common  af 
fairs  ;  but  in  philosophical  disquisitions,  we  ought  to  abstract  from  our 
senses,  and  consider  things  themselves,  distinct  from  what  are  only  sensible 
measures  of  them.  For  it  may  be  that  there  is  no  body  really  at  rest,  to 
which  the  places  and  motions  of  others  may  be  referred. 

But  we  may  distinguish  rest  and  motion,  absolute  and  relative,  one  from 
the  other  by  their  properties,  causes  and  effects.  It  is  a  property  of  rest, 
that  bodies  really  at  rest  do  rest  in  respect  to  one  another.  And  therefore 
as  it  is  possible,  that  in  the  remote  regions  of  the  fixed  stars,  or  perhaps 
far  beyond  them,  there  may  be  some  body  absolutely  at  rest ;  but  impossi 
ble  to  know,  from  the  position  of  bodies  to  one  another  in  our  regions 
whether  any  of  these  do  keep  the  same  position  to  that  remote  body;  it 
follows  that  absolute  rest  cannot  be  determined  from  the  position  of  bodies 
in  our  regions. 

It  is  a  property  of  motion,  that  the  parts,  which  retain  given  positions 
to  their  wholes,  do  partake  of  the  motions  of  those  wholes.  For  all  the 
parts  of  revolving  bodies  endeavour  to  recede  from  the  axis  of  motion  ; 
and  the  impetus  of  bodies  moving  forward,  arises  from  the  joint  impetus 
of  all  the  parts.  Therefore,  if  surrounding  bodies  are  moved,  those  that 
are  relatively  at  rest  within  them,  will  partake  of  their  motion.  Upon 
which  account,  the  true  and  absolute  motion  of  a  body  cannot  be  Jeter- 


8C  THE    MATHEMATICAL    PRINCIPLES 

mined  by  the  translation  of  it  from  those  which  only  seem  to  rest ;  for  the 
external  bodies  ought  not  only  to  appear  at  rest,  but  to  be  really  at  rest. 
For  otherwise,  all  included  bodies,  beside  their  translation  from  near  the 
surrounding  ones,  partake  likewise  of  their  true  motions ;  and  though  that 
translation  were  not  made  they  would  not  be  really  at  rest,  but  only  seem 
to  be  so.  For  the  surrounding  bodies  stand  in  the  like  relation  to  the 
surrounded  as  the  exterior  part  of  a  whole  does  to  the  interior,  or  as  the 
shell  does  to  the  kernel ;  but,  if  the  shell  moves,  the  kernel  will  also 
move,  as  being  part  of  the  whole,  without  any  removal  from  near  the  shell. 

A  property,  near  akin  to  the  preceding,  is  this,  that  if  a  place  is  moved, 
whatever  is  placed  therein  moves  along  with  it ;  and  therefore  a  body, 
which  is  moved  from  a  place  in  motion,  partakes  also  of  the  motion  of  its 
place.  Upon  which  account,  all  motions,  from  places  in  motion,  are  no 
other  than  parts  of  entire  and  absolute  motions ;  and  every  entire  motion 
is  composed  of  the  motion  of  the  body  out  of  its  first  place,  and  the 
motion  of  this  place  out  of  its  place ;  and  so  on,  until  we  come  to  some 
immovable  place,  as  in  the  before-mentioned  example  of  the  sailor.  Where 
fore,  entire  and  absolute  motions  can  be  no  otherwise  determined  than  by 
immovable  places :  and  for  that  reason  I  did  before  refer  those  absolute 
motions  to  immovable  places,  but  relative  ones  to  movable  places.  Now 
no  other  places  are  immovable  but  those  that,  from  infinity  to  infinity,  do 
all  retain  the  same  given  position  one  to  another ;  and  upon  this  account 
must  ever  remain  unmoved ;  and  do  thereby  constitute  immovable  space. 

The  causes  by  which  true  and  relative  motions  are  distinguished,  one 
from  the  other,  are  the  forces  impressed  upon  bodies  to  generate  motion. 
True  motion  is  neither  generated  nor  altered,  but  by  some  force  impressed 
upon  the  body  moved :  but  relative  motion  may  be  generated  or  altered 
without  any  force  impressed  upon  the  body.  For  it  is  sufficient  only  to 
impress  some  force  on  other  bodies  with  which  the  former  is  compared, 
that  by  their  giving  way,  that  relation  may  be  changed,  in  which  the  re 
lative  rest  or  motion  of  this  other  body  did  consist.  Again,  true  motion 
suffers  always  some  change  from  any  force  impressed  upon  the  moving 
body  ;  but  relative  motion  docs  not  necessarily  undergo  any  change  by  such 
forces.  For  if  the  same  forces  are  likewise  impressed  on  those  other  bodies, 
with  which  the  comparison  is  made,  that  the  relative  position  may  be  pre 
served,  then  that  condition  will  be  preserved  in  which  the  relative  motion 
consists.  And  therefore  any  relative  motion  may  be  changed  when  the 
true  motion  remains  unaltered,  and  the  relative  may  be  preserved  when  the 
true  suffers  some  change.  Upon  which  accounts;  true  motion  does  by  no 
means  consist  in  such  relations. 

The  effects  whicli  distinguish  absolute  from  relative  motion  arc,  the 
forces  of  receding  from  the  axis  of  circular  motion.  For  there  are  no  such 
forces  in  a  circular  motion  purely  relative,  but  in  a  true  and  absolute  cir 
cular  motion.,  they  are  greater  or  less,  according  t »  the  quantity  of  the 


OF    NATURAL    PHILOSOPHY.  «1 

motion.  If  a  vessel,  hung:  by  &  }ong  cord,  is  so  often  turned  ubout  that  the 
cord  is  strongly  twisted,  then  filled  with  water,  and  held  at  rest  together 
with  the  water ;  after,  by  the  sudden  action  of  another  force,  it  is  whirled 
about  the  contrary  way,  and  while  the  cord  is  untwisting  itself,  the  vessel 
continues  for  some  time  in  this  motion ;  the  surface  of  the  water  will  at 
first  be  plain,  as  before  the  vessel  began  to  move :  but  the  vessel;  by  grad 
ually  communicating  its  motion  to  the  water,  will  make  it  begin  sensibly 
^to  revolve,  and  recede  by  little  and  little  from  the  middle,  and  ascend  to  the 
sides  of  the  vessel,  forming  itself  into  a  concave  figure  (as  I  have  experi 
enced),  and  the  swifter  the  motion  becomes,  the  higher  will  the  water  rise, 
till  at  last,  performing  its  revolutions  in  the  same  times  with  the  vessel, 
it  becomes  relatively  at  rest  in  it.  This  ascent  of  the  water  shows  its  en 
deavour  to  recede  from  the  axis  of  its  motion ;  and  the  true  and  absolute 
circular  motion  of  the  water,  which  is  here  directly  contrary  to  the  rela- 
tivej  discovers  itself,  and  may  be  measured  by  this  endeavour.  At  first, 
when  the  relative  motion  of  the  water  in  the  vessel  was  greatest,  it  pro 
duced  no  endeavour  to  recede  from  the  axis ;  the  water  showed  no  tendency 
to  the  circumference,  nor  any  ascent  towards  the  sides  of  the  vessel,  but 
remained  of  a  plain  surface,  and  therefore  its  true  circular  motion  had  not 
yet  begun.  But  afterwards,  when  the  relative  motion  of  the  water  had 
decreased,  the  ascent  thereof  towards  the  sides  of  the  vessel  proved  its  en 
deavour  to  recede  from  the  axis ;  and  this  endeavour  showed  the  real  cir 
cular  motion  of  the  water  perpetually  increasing,  till  it  had  acquired  its 
greatest  quantity,  wh en  the  water  rested  relatively  in  the  vessel.  And 
therefore  this  endeavour  does  not  depend  upon  any  translation  of  the  water 
in  respect  of  the  ambient  bodies,  nor  can  true  circular  motion  be  defined 
by  such  translation.  There  is  only  one  real  circular  motion  of  any  one 
revolving  body,  corresponding  to  only  one  power  of  endeavouring  to  recede 
from  its  axis  of  motion,  as  its  proper  and  adequate  effect ;  but  relative 
motions,  in  one  and  the  same  body,  are  innumerable,  according  to  the  various 
relations  it  bears  to  external  bodies,  and  like  other  relations,  arc  altogether 
destitute  of  any  real  effect,  any  otherwise  than  they  may  perhaps  par 
take  of  that  one  only  true  motion.  And  therefore  in  their  system  who 
suppose  that  our  heavens,  revolving  below  the  sphere  of  the  fixed  stars, 
carry  the  planets  along  with  them ;  the  several  parts  of  those  heavens,  and 
the  planets,  which  are  indeed  relatively  at  rest  in  their  heavens,  do  yet 
really  move.  For  they  change  their  position  one  to  another  (which  never 
happens  to  bodies  truly  at  rest),  and  being  carried  together  with  their 
heavens,  partake  of  their  motions,  and  as  parts  of  revolving  wholes, 
endeavour  to  recede  from  the  axis  of  their  motions. 

Wherefore  relative  quantities  are  not  the  quantities  themselves,  whose 
names  they  bear,  but  those  sensible  measures  of  them  (either  accurate  cr 
inaccurate),  which  arc  commonly  used  instead  of  the  measured  quantities 
themselves.  And  if  the  meaning  of  words  is  to  he  determined  bv  their 


82  THE  MATHEMATICAL    PRINCIPLES 

use,  then  by  the  names  time,  space,  place  and  motion,  their  measures  arv 
properly  to  be  understood ;  and  the  expression  will  be  unusual,  and  purely 
mathematical,  if  the  measured  quantities  themselves  are  meant.  Upon 
which  account,  they  do  strain  the  sacred  writings,  who  there  interpret 
those  words  for  the  measured  quantities.  Nor  do  those  less  defile  the 
purity  of  mathematical  and  philosophical  truths,  who  confound  real  quan 
tities  themselves  with  their  relations  and  vulgar  measures. 

It  is  indeed  a  matter  of  great  difficulty  to  discover,  and  effectually  to 
distinguish,  the  true  motions  of  particular  bodies  from  the  apparent ;  be 
cause  the  parts  of  that  immovable  space,  in  which  those  motions  are  per 
formed,  do  by  no  means  come  under  the  observation  of  our  senses.  Yet 
the  thing  is  not  altogether  desperate :  for  we  have  some  arguments  to 
guide  us,  partly  from  the  apparent  motions,  which  are  the  differences  of 
the  true  motions ;  partly  from  the  forces,  which  are  the  causes  and  effects 
of  the  true  motions.  For  instance,  if  tAvo  globes,  kept  at  a  given  distance 
one  from  the  other  by  means  of  a  cord  that  connects  them,  were  revolved 
about  their  common  centre  of  gravity,  we  might,  from  the  tension  of  the 
cord,  discover  the  endeavour  of  the  globes  to  recede  from  the  axis  of  their 
motion,  and  from  thence  we  might  compute  the  quantity  of  their  circular 
motions.  And  then  if  any  equal  forces  should  be  impressed  at  once  on  the 
alternate  faces  of  the  globes  to  augment  or  diminish  their  circular  motions, 
from  the  increase  or  decr»  ase  of  the  tensicn  of  1  le  cord,  we  might  infer 
the  increment  or  decrement  of  their  motions :  and  thence  would  be  found 
on  what  faces  those  forces  ought  to  be  impressed,  that  the  motions  of  the 
globes  might  be  most  augmented  ;  that  is,  we  might  discover  their  hinder- 
most  faces,  or  those  which,  in  the  circular  motion,  do  follow.  But  the 
faces  which  follow  being  known,  and  consequently  the  opposite  ones  that 
precede,  we  should  likewise  know  the  determination  of  their  motions.  And 
thus  we  might  find  both  the  quantity  and  the  determination  of  this  circu 
lar  motion,  even  in  an  immense  vacuum,  where  there  was  nothing  external 
or  sensible  with  which  the  globes  could  be  compared.  But  now,  if  in  that 
space  some  remote  bodies  were  placed  that  kept  always  a  given  position 
one  to  another,  as  the  fixed  stars  do  in  our  regions,  we  could  not  indeed 
determine  from  the  relative  translation  of  the  globes  among  those  bodies, 
whether  the  motion  did  belong  to  the  globes  or  to  the  bodies.  But  if  we 
observed  the  cord,  and  found  that  its  tension  was  that  very  tension  which 
the  motions  of  the  globes  required,  we  might  conclude  the  motion  to  be  in 
the  globes,  and  the  bodies  to  be  at  rest ;  and  then,  lastly,  from  the  trans 
lation  of  the  globes  among  the  bodies,  we  should  find  the  determination  oi 
their  motions.  But  how  we  are  to  collect  the  true  motions  from  their 
causes,  effects,  and  apparent  differences ;  and,  vice  versa,  how  from  the  mo 
tions,  either  true  or  apparent,  we  may  come  to  the  knowledge  of  theii 
causes  and  effects,  shall  be  explained  more  at  large  in  the  following  tra<;t 
For  to  this  end  it  was  that  I  composed  it. 


OF    NATURAL    PHILOSOPHY. 


AXIOMS,  OR  LAWS  OF  MOTION. 

LAW  I. 

Hvery  body  perseveres  in  its  state  of  rest,  or  of  uniform  motion  in  a  ri^ht 
line,  unless  it  is  compelled  to  change  that  state  by  forces  impressed 
thereon. 

PROJECTILES  persevere  in  their  motions,  so  far  as  they  are  not  retarded 
by  the  resistance  of  the  air,  or  impelled  downwards  by  the  force  of  gravity 
A  top,  whose  parts  by  their  cohesion  are  perpetually  drawn  aside  from 
rectilinear  motions,  does  not  cease  its  rotation,  otherwise  than  as  it  is  re 
tarded  by  the  air.  The  greater  bodies  of  the  planets  and  comets,  meeting 
with  less  resistance  in  more  free  spaces,  preserve  then  jDotions  both  pro 
gressive  and  circular  for  a  much  longer  time. 

LAW   II. 

The  alteration  of  motion  is  ever  proportional  to  the  motive  force  imp  reus 
ed  ;  and  is  made  in  the  direction  of  the  right  line  in.  which  that  force 
is  impressed. 

If  any  force  generates  a  motion,  a  double  force  will  generate  double  the 
motion,  a  triple  force  triple  the  motion,  whether  that  force  be  impressed 
altogether  and  at  once,  or  gradually  and  successively.  And  this  motion 
(being  always  directed  the  same  way  with  the  generating  force),  if  the  body 
moved  before,  is  added  to  or  subducted  from  the  former  motion,  according 
as  they  directly  conspire  with  or  are  directly  contrary  to  each  other ;  or 
obliquely  joined,  when  they  are  oblique,  so  as  to  produce  a  new  motion 
compounded  from  the  determination  of  both. 

LAW  III. 

To  every  action  there  is  always  opposed  an  equal  reaction :  or  the  mu 
tual  actions  of  two  bodies  upon  each  other  are  always  equal,  and  di 
rected  to  contrary  parts. 

Whatever  draws  or  presses  another  is  as  much  drawn  or  pressed  by  that 
other.  If  you  press  a  stone  with  your  finger,  the  finger  is  also  pressed  by 
the  stone.  If  a  horse  draws  a  stone  tied  to  a  rope,  the  horse  (if  I  may  so 
say)  will  be  equally  drawn  back  towards  the  stone:  for  the  distended  rope, 
by  the  same  endeavour  to  relax  or  unbend  itself,  will  draw  the  horse  as 
much  towards  the  stone,  as  it  does  the  stone  towards  the  horse,  and  will 
obstruct  the  progress  of  the  one  as  much  as  it  advances  that  of  the  other. 


84  THE    MATHEMATICAL    PRINCIPLES 

If  a  body  impinge  upon  another,  and  by  its  force  change  the  motion  of  (It* 
other,  that  body  also  (because  of  the  equality  of  the  mutual  pressure)  will 
undergo  an  equal  change,  in  its  own  motion,  towards  the  contrary  part. 
The  changes  made  by  these  actions  are  equal,  not  in  the  velocities  but  in 
the  motions  of  bodies ;  that  is  to  say,  if  the  bodies  are  not  hindered  by  any 
other  impediments.  For,  because  the  motions  are  equally  changed,  the 
changes  of  the  velocities  made  towards  contrary  parts  are  reciprocally  pro 
portional  to  the  bodies.  This  law  takes  place  also  in  attractions,  as  will 
be  proved  in  the  next  scholium. 

COROLLARY  I. 

A  body  by  two  forces  conjoined  will  describe  the  diagonal  of  a  parallelo 
gram,  in  the  same  time  that  it  wovld  describe  the  sides,  by  those  forces 
apart. 

If  a  body  in  a  given  time,  by  the  force  M  impressed 
apart  in  the  place  A,  should  with  an  uniform  motion        / 
be  carried  from  A  to  B ;  and  by  the  force  N  impressed 
apart  in  the  same  place,  should  be  carried  from  A  to  c ~\) 
C  ;  complete  the  parallelogram  ABCD,  and,  by  both  forces  acting  together, 
it  will  in  the  same  time  be  carried  in  the  diagonal  from  A  to  D.     For 
since  the  force  N  acts  in  the  direction  of  the  line  AC,  parallel  to  BD,  this 
force  (by  the  second  law)  will  not  at  all  alter  the  velocity  generated  by  the 
other  force  M,  by  which  the  body  is  carried  towards  the  line  BD.     The 
body  therefore  will  arrive  at  the  line  BD  in  the  same  time,  whether  the 
rorce  N  be  impressed  or  not ;  and  therefore  at  the  end  of  that  time  it  will 
he  found  somewhere  in  the  line  BD.     By  the  same  argument,  at  the  end 
of  the  same  time  it  AY  ill  be  found  somewhere  in  the  line  CD.     Therefore  it 
will  be  found  in  the  point  D,  where  both  lines  meet.     But  it  will  move  in 
;i  right  line  from  A  to  D,  by  Law  I. 

COROLLARY  II. 

And  hence  is  explained  the  composition  of  any  one  direct  force  AD,  out 
of  any  two  oblique  forces  AC  and  CD ;  and,  on  the  contrary,  the  re 
solution  of  any  one  direct  force  AD  into  two  oblique  forces  AC  and 
CD :  which  composition  and  resolution  are  abundantly  confirmed  from, 
mechanics. 

As  if  the  unequal  radii  OM  and  ON  drawn  from  the  centre  O  of  any 
wheel,  should  sustain  the  weights  A  and  P  by  the  cords  MA  and  NP ;  and 
the  forces  of  those  weights  to  move  the  wheel  were  required.  Through  the 
rentre  O  draw  the  right  line  KOL,  meeting  the  cords  perpendicularly  in 
A  and  L;  and  from  the  centre  O,  with  OL  the  greater  of  the  distances 


OF    NATURAL    PHILOSOPHY. 

OK  arid  OL,  describe  a  circle,  meeting  the  cord 
MA  in  D  :  and  drawing  OD,  make  AC  paral-  "^ 
lei  and  DC  perpendicular  thereto.  Now,  it 
being  indifferent  whether  the  points  K,  L,  D,  of 
the  cords  be  lixed  to  the  plane  of  the  wheel  or 
not,  the  weights  will  have  the  same  effect 
whether  they  are  suspended  from  the  points  K 
and  L,  or  from  D  and  L.  Let  the  whole  force 
of  the  weight  A  be  represented  by  the  line  AD, 
and  let  it  be  resolved  into  the  forces  AC  and 
CD  ;  of  which  the  force  AC,  drawing  the  radius 
OD  directly  from  the  centre,  will  have  no  effect  to  move  the  wheel :  but 
the  other  force  DC,  drawing  the  radius  DO  perpendicularly,  will  have  the 
same  effect  as  if  it  drew  perpendicularly  the  radius  OL  equal  to  OD  ;  that 
is,  it  w  ill  have  the  same  effect  as  the  weight  P,  if  that  weight  is  to  the 
weight  A  as  the  force  DC  is  to  the  force  DA  ;  that  is  (because  of  the  sim 
ilar  triangles  ADC,  DOK),  as  OK  to  OD  or  OL.  Therefore  the  weights  A 
and  P,  which  are  reciprocally  as  the  radii  OK  and  OL  that  lie  in  the  same 
right  line,  will  be  equipollent,  and  so  remain  in  equilibrio  ;  which  is  the  well 
known  property  of  the  balance,  the  lever,  and  the  wheel.  If  either  weight  is 
greater  than  in  this  ratio,  its  force  to  move  the  wheel  will  be  so  much  greater. 

If  the  weight  p,  equal  to  the  weight  P,  is  partly  suspended  by  the 
cord  NJO,  partly  sustained  by  the  oblique  plane  pG ;  draw  p}i,  NH,  the 
former  perpendicular  to  the  horizon,  the  latter  to  the  plane  pG  ;  and  if 
the  force  of  the  weight  p  tending  downwards  is  represented  by  the  line 
/?H,  it  may  be  resolved  into  the  forces  joN,  HN.  If  there  was  any  plane 
/?Q,  perpendicular  to  the  cord  y?N,  cutting  the  other  plane  pG  in  a  line 
parallel  to  the  horizon,  and  the  weight  p  was  supported  only  by  those 
planes  pQ,  pG,  it  would  press  those  planes  perpendicularly  with  the  forces 
pN,  HN;  to  wit,  the  plane  joQ,  with  the  force  joN,  and  the  plane  pG  with 
the  force  HN.  And  therefore  if  the  plane  pQ  was  taken  away,  so  thnt 
the  weight  might  stretch  the  cord,  because  the  cord,  now  sustaining  the 
weight,  supplies  the  place  of  the  plane  that  was  removed,  it  will  be  strained 
by  the  same  force  joN  which  pressed  upon  the  plane  before.  Therefore, 
the  tension  of  this  oblique  cord  joN  will  be  to  that  of  the  other  perpendic 
ular  cord  PN  as  jt?N  to  joH.  And  therefore  if  the  weight  p  is  to  the 
weight  A  in  a  ratio  compounded  of  the  reciprocal  ratio  of  the  least  distances 
of  the  cords  PN,  AM,  from  the  centre  of  the  wheel,  and  of  the  direct  ratio  of 
pH  tojoN,  the  weights  will  have  the  same  effect  towards  moving  the  wheel, 
and  will  therefore  sustain  each  other :  as  any  one  may  find  by  experiment. 

But  the  weight  p  pressing  upon  those  two  oblique  planes,  may  be  con 
sidered  as  a  wedge  between  the  two  internal  surfaces  of  a  body  split  by  it; 
and  hence  tlif  ft  IV.P*  of  th^  v, ^dge  and  the  mallet  may  be  determined;  foi 


8G  THE    MATHEMATICAL    PRINCIPLES 

because  the  force  with  which  the  weight  p  presses  the  plane  pQi  is  to  the 
force  with  which  the  same,  whether  by  its  own  gravity,  or  by  the  blow  of 
a  mallet,  is  impelled  in  the  direction  of  the  line  joH  towards  both  the 
planes,  as  joN  to  pH ;  and  to  the  force  with  which  it  presses  the  other 
plane  pG,  as  joN  to  NH.  And  thus  the  force  of  the  screw  may  be  deduced 
from  a  like  resolution  of  forces ;  it  being  no  other  than  a  wedge  impelled 
with  the  force  of  a  lever.  Therefore  the  use  of  this  Corollary  spreads  far 
and  wide,  and  by  that  diffusive  extent  the  truth  thereof  is  farther  con 
firmed.  For  on  what  has  been  said  depends  the  whole  doctrine  of  mechan 
ics  variously  demonstrated  by  different  authors.  For  from  hence  are  easily 
deduced  the  forces  of  machines,  which  are  compounded  of  wheels,  pullics, 
levers,  cords,  and  weights,  ascending  directly  or  obliquely,  and  other  mechan 
ical  powers ;  as  also  the  force  of  the  tendons  to  move  the  bones  of  animals. 

COROLLARY  III. 

The  (/uaittity  of  motion,  which  is  collected  by  taking  the  sum  of  the  mo 
tions  directed  towards  the  same  parts,  and  the  difference  of  those  that 
are  directed  to  contrary  parts,  suffers  no  change  from  the  action  oj 
bodies  among  themselves. 

For  action  and  its  opposite  re-action  are  equal,  by  Law  III,  and  there 
fore,  by  Law  II,  they  produce  in  the  motions  equal  changes  towards  oppo 
site  parts.  Therefore  if  the  motions  are  directed  towards  the  same  parts. 
whatever  is  added  to  the  motion  of  the  preceding  body  will  be  subducted 
from  the  motion  of  that  which  follows  ;  so  that  the  sum  will  be  the  same 
as  before.  If  the  bodies  meet,  with  contrary  motions,  there  will  be  an 
equal  deduction  from  the  motions  of  both ;  and  therefore  the  difference  of 
the  motions  directed  towards  opposite  parts  will  remain  the  same. 

Thus  if  a  spherical  body  A  with  two  parts  of  velocity  is  triple  of  a 
spherical  body  B  which  follows  in  the  same  right  line  with  ten  parts  of 
velocity,  the  motion  of  A  will  be  to  that  of  B  as  6  to  10.  Suppose, 
then,  their  motions  to  be  of  6  parts  and  of  10  parts,  and  the  sum  will  be 
16  parts.  Therefore,  upon  the  meeting  of  the  bodies,  if  A  acquire  3,  4, 
or  5  parts  of  motion,  B  will  lose  as  many ;  and  therefore  after  reflexion 
A  will  proceed  With  9,  10,  or  11  parts,  and  B  with  7,  6,  or  5  parts;  the 
sum  remaining  always  of  16  parts  as  before.  If  the  body  A  acquire  9, 
10,  11,  or  12  parts  of  motion,  and  therefore  after  meeting  proceed  with 
15,  16,  17,  or  18  parts,  the  body  B,  losing  so  many  parts  as  A  has  got, 
will  either  proceed  with  1  part,  having  lost  9,  or  stop  and  remain  at  rest, 
as  having  lost  its  whole  progressive  motion  of  10  parts ;  or  it  will  go  back 
with  1  part,  having  not  only  lost  its  whole  motion,  but  (if  1  may  so  say) 
one  part  more;  or  it  will  go  back  with  2  parts,  because  a  progressive  mo 
tion  of  12  parts  is  taken  off.  And  so  the  sums  of  the  Conspiring  motions 
15  ,1,  or  16-1-0,  and  the  differences  of  the  contrary  i  otions  17 — 1  and 


OF    NATURAL    PHILOSOPHY. 

[S — 2,  will  always  be  equal  to  16  parts,  as  they  were  before  tie  meeting 
and  reflexion  of  the  bodies.  But,  the  motions  being  known  with  whicli 
the  bodies  proceed  after  reflexion,  the  velocity  of  either  will  be  also  known, 
by  taking  the  velocity  after  to  the  velocity  before  reflexion,  as  the  motion 
after  is  to  the  motion  before.  As  in  the  last  case,  where  the  motion  of  tho 
body  A  was  of  0  parts  before  reflexion  and  of  IS  parts  after,  and  the 
velocity  was  of  2  parts  before  reflexion,  the  velocity  thereof  after  reflexion 
will  be  found  to  be  of  6  parts ;  by  saying,  as  the  0  parts  of  motion  before 
to  18  parts  after,  so  are  2  parts  of  velocity  before  reflexion  to  (5  parts  after. 
But  if  the  bodies  are  cither  not  spherical,  or,  moving  in  different  right 
lines,  impinge  obliquely  one  upon  the  other,  and  their  mot'ons  after  re 
flexion  are  required,  in  those  cases  we  are  first  to  determine  the  position 
of  the  plane  that  touches  the  concurring  bodies  in  the  point  of  concourse  , 
then  the  motion  of  each  body  (by  Corol.  II)  is  to  be  resolved  into  two,  one 
perpendicular  to  that  plane,  and  the  other  parallel  to  it.  This  done,  be 
cause  the  bodies  act  upon  each  other  in  the  direction  of  a  line  perpendicu 
lar  to  this  plane,  the  parallel  motions  are  to  be  retained  the  same  after 
reflexion  as  before ;  and  to  the  perpendicular  motions  we  are  to  assign 
equal  changes  towards  the  contrary  parts ;  in  such  manner  that  the  sum 
of  the  conspiring  and  the  difference  of  the  contrary  motions  may  remain 
the  same  as  before.  From  such  kind  of  reflexions  also  sometimes  arise 
the  circular  motions  of  bodies  about  their  own  centres.  But  these  are 
cases  which  I  do  not  consider  in  what  follows ;  and  it  would  be  too  tedious 
to  demonstrate  every  particular  that  relates  to  this  subject. 

COROLLARY  IV. 

The  common  centre  of  gravity  of  two  or  more  bodies  does  not  alter  its 
state  of  'motion  or  rest  by  the  actions  of  the  bodies  among  themselves ; 
and  therefore  the  common  centre  of  gravity  of  all  bodies  acting  upon 
each  other  (excluding  outward  actions  and  impediments)  is  either  at 
rest,  or  moves  uniformly  in  a  right  line. 

For  if  two  points  proceed  with  an  uniform  motion  in  right  lines,  and 
their  distance  be  divided  in  a  given  ratio,  the  dividing  point  will  be  either 
at  rest,  or  proceed  uniformly  in  a  right  line.  This  is  demonstrated  here 
after  in  Lem.  XXIII  and  its  Corol.,  when  the  points  are  moved  in  the  same 
plane ;  and  by  a  like  way  of  arguing,  it  may  be  demonstrated  when  the 
points  are  not  moved  in  the  same  plane.  Therefore  if  any  number  of 
Kdies  move  uniformly  in  right  lines,  the  common  centre  of  gravity  of  any 
two  of  them  is  either  at  rest,  or  proceeds  uniformly  in  a  right  line ;  because 
the  line  which  connects  the  centres  of  those  two  bodies  so  moving  is  divided  at 
that  common  centre  in  a  given  ratio.  In  like  manner  the  common  centre 
of  those  two  and  that  of  a  third  body  will  be  either  at  rest  or  moving  uni 
formly  in  aright  line  because  at  that  centre  the  distance  1  etween  th«? 


88  THE    MATHEMATICAL    PRINCIPLES 

common  centre  of  the  two  bodies,  and  the  centre  of  this  last,  is  divided  in 
a  given  ratio.  In  like  manner  the  common  centre  of  these  three,  and  of  a 
fourth  body,  is  either  at  rest,  or  moves  uniformly  in  a  right  line ;  because 
the  distance  between  the  common  centre  of  the  three  bodies,  and  the  centre 
of  the  fourth  is  there  also  divided  in  a  given  ratio,  and  so  on  m  itifinitum. 
Therefore,  in  a  system  of  bodies  where  there  is  neither  any  mutual  action 
among  themselves,  nor  any  foreign  force  impressed  upon  them  from  without, 
and  which  consequently  move  uniformly  in  right  lines,  the  common  centre  of 
gravity  of  them  all  is  either  at  rest  or  moves  uniformly  forward  in  a  right  line. 
Moreover,  in  a  system  of  two  bodies  mutually  acting  upon  each  other, 
since  the  distances  between  their  centres  and  the  common  centre  of  gravity 
of  both  are  reciprocally  as  the  bodies,  the  relative  motions  of  those  bodies, 
whether  of  approaching  to  or  of  receding  from  that  centre,  will  be  equal 
among  themselves.  Therefore  since  the  changes  which  happen  to  motions 
are  equal  and  directed  to  contrary  parts,  the  common  centre  of  those  bodies, 
by  their  mutual  action  between  themselves,  is  neither  promoted  nor  re 
tarded,  nor  suffers  any  change  as  to  its  state  of  motion  or  rest.  But  in  a 
system  of  several  bodies,  because  the  common  centre  of  gravity  of  any  two 
acting  mutually  upon  each  other  suffers  no  change  in  its  state  by  that  ac 
tion  :  and  much  less  the  common  centre  of  gravity  of  the  others  with  which 
that  action  does  not  intervene ;  but  the  distance  between  those  two  centres 
is  divided  by  the  common  centre  of  gravity  of  all  the  bodies  into  parts  re 
ciprocally  proportional  to  the  total  sums  of  those  bodies  whose  centres  they 
are :  and  therefore  while  those  two  centres  retain  their  state  of  motion  or 
rest,  xhe  common  centre  of  all  does  also  retain  its  state :  it  is  manifest  that 
the  common  centre  of  all  never  suffers  any  change  in  the  state  of  its  mo 
tion  or  rest  from  the  actions  of  any  two  bodies  between  themselves.  But 
in  such  &  system  all  the  actions  of  the  bodies  among  themselves  either  hap 
pen  between  two  bodies,  or  are  composed  of  actions  interchanged  between 
some  two  bodies ;  and  therefore  they  do  never  produce  any  alteration  in 
the  comrrv  n  centre  of  alias  to  its  state  of  motion  or  rest.  Wherefore 
tiince  that  centre,  when  the  bodies  do  not  act  mutually  one  upon  another, 
Oilier  is  nt  rest  or  moves  uniformly  forward  in  some  right  line,  it  will, 
:v\>U7ithst?nding  the  mutual  actions  of  the  bodies  among  themselves,  always 
jAY-jevere  in  its  state,  either  of  rest,  or  of  proceeding  uniformly  in  a  right 
liiv,,  unless  it  is  forced  out  of  this  state  by  the  action  of  some  power  im- 
prev^-d  from  without  upon  the  whole  system.  And  therefore  the  same  law 
take*1  place  in  a  system  consisting  of  many  bodies  as  in  one  single  body, 
with  wsgard  to  their  persevering  in  their  state  of  motion  or  of  rest.  For 
the  pi  \\jressive  motion,  whether  of  one  single  body,  or  of  a  whole  system  of 
bodies  us  always  to  be  estimated  from  the  motion  of  the  centre  of  gravity. 

COROLLARY  V. 

The  motions  cf  bcdies  included  in  a  given  space  a  ~e  Ike  same  among 


OF    NATURAL    PHILOSOPHY.  89 

themselves,  whether  that  space  is  at  rest,  or  moves  uniformly  forwards 

in  a  right  line  without  any  circular  motion. 

For  the  differences  of  the  motions  tending  towards  the  same  parts,  and 
the  sums  of  those  that  tend  towards  contrary  parts,  are,  at  first  (by  sup 
position),  in  both  cases  the  same ;  and  it  is  from  those  sums  and  differences 
that  the  collisions  and  impulses  do  arise  with  which  the  bodies  mutually 
impinge  one  upon  another.  Wherefore  (by  Law  II),  the  effects  of  those 
collisions  will  be  equal  in  both  cases ;  and  therefore  the  mutual  motions 
of  the  bodies  among  themselves  in  the  one  case  will  remain  equal  to  the 
mutual  motions  of  the  bodies  among  themselves  in  the  other.  A  clear 
proof  of  which  we  have  from  the  experiment  of  a  ship  ;  where  all  motions 
happen  after  the  same  manner,  whether  the  ship  is  at  rest,  or  is  carried 
uniformly  forwards  in  a  right  line. 

COROLLARY  VI. 

If  bodies,  any  how  moved  among  themselves,  are  urged  in  the  direct-ton 
of  parallel  lines  by  equal  accelerative  forces,  they  will  all  continue  to 
move  among  themselves,  after  the  same  manner  as  if  they  had  been 
urged  by  no  such  forces. 

For  these  forces  acting  equally  (with  respect  to  the  quantities  of  the 
DO  dies  to  be  moved),  and  in  the  direction  of  parallel  lines,  will  (by  Law  II) 
move  all  the  bodies  equally  (as  to  velocity),  and  therefore  will  never  pro 
duce  any  change  in  the  positions  or  motions  of  the  bodies  among  themselves. 

SCHOLIUM. 

Hitherto  I  have  laid  down  such  principles  as  have  been  received  by  math 
ematicians,  and  are  confirmed  by  abundance  of  experiments.  By  the  first 
two  Laws  and  the  first  two  Corollaries,  Galileo  discovered  that  the  de 
scent  of  bodies  observed  the  duplicate  ratio  of  the  time,  and  that  the  mo 
tion  of  projectiles  was  in  the  curve  of  a  parabola;  experience  agreeing 
with  both,  unless  so  far  as  these  motions  are  a  little  retarded  by  the  re 
sistance  of  the  air.  When  a  body  is  falling,  the  uniform  force  of  its 
gravity  acting  equally,  impresses,  in  equal  particles  of  time,  equal  forces 
upon  that  body,  and  therefore  generates  equal  velocities;  and  in  the  whole 
time  impresses  a  whole  force,  and  generates  a  whole  velocity  proportional 
to  the  time.  And  the  spaces  described  in  proportional  times  are  as  the 
velocities  and  the  times  conjunctly ;  that  is,  in  a  duplicate  ratio  of  the 
times.  And  when  a  body  is  thrown  upwards,  its  uniform  gravity  im 
presses  forces  and  takes  off  velocities  proportional  to  the  times ;  and  the 
times  of  ascending  to  the  greatest  heights  are  as  the  velocities  to  be  taken 
off,  and  those  heights  are  as  the  velocities  and  the  times  conjunetly,  or  ir, 
the  duplicate  ratio  of  the  velocities.  And  if  a  body  be  projected  in  any 
direction,  the  motion  arising  from  its  projection  jS  compounded  with  the 


90 


THE    MATHEMATICAL    PRINCIPLES 


motion  arising  from  its  gravity.  As  if  the  body  A  by  its  motion  of  pio- 
jection  alone  could  describe  in  a  given  time  the  right  line 
AB,  and  with  its  motion  of  falling  alone  could  describe  in 
the  same  time  the  altitude  AC  ;  complete  the  paralello- 
gram  ABDC,  and  the  body  by  that  compounded  motion 
will  at  the  end  of  the  time  be  found  in  the  place  D  ;  and 
the  curve  line  AED,  which  that  body  describes,  will  be  a 
parabola,  to  which  the  right  line  AB  will  be  a  tangent  in 
A  ;  and  whose  ordinate  BD  will  be  as  the  square  of  the  line  AB.  On  the 
same  Laws  and  Corollaries  depend  those  things  which  have  been  demon 
strated  concerning  the  times  of  the  vibration  of  pendulums,  and  are  con 
firmed  by  the  daily  experiments  of  pendulum  clocks.  By  the  same,  to 
gether  with  the  third  Law,  Sir  Christ.  Wren,  Dr.  Wallis,  and  Mr.  Huv- 
gens,  the  greatest  geometers  of  our  times,  did  severally  determine  the  rules 
of  the  congress  and  reflexion  of  hard  bodies,  and  much  about  the  same 
time  communicated  their  discoveries  to  the  Royal  Society,  exactly  agreeing 
among  themselves  as  to  those  rules.  Dr.  Wallis,  indeed,  was  something 
more  early  in  the  publication  ;  then  followed  Sir  Christopher  Wren,  and, 
lastly,  Mr.  Huygens.  But  Sir  Christopher  Wren  confirmed  the  truth  of 
the  thing  before  the  Royal  Society  by  the  experiment  of  pendulums,  which 
Mr.  Mariottc  soon  after  thought  fit  to  explain  in  a  treatise  entirely  upon 
that  subject.  But  to  bring  this  experiment  to  an  accurate  agreement  with 
the  theory,  we  are  to  have  a  due  regard  as  well  to  the  resistance  of  the  air 


bodies. 


Let  the  spherical  bodies 
CD  F    II 


as  to  the  clastic  force  of  the  concurrin 
A,  B  be  suspended  by  the  parallel  and 
equal  strings  AC,  Bl),  from  the  centres 
C,  D.  About  these  centres,  with  those 
intervals,  describe  the  semicircles  EAF, 
GBH,  bisected  by  the  radii  CA,  DB. 
Bring  the  body  A  to  any  point  R  of  the 
arc  EAF,  and  (withdrawing  the  body 
B)  let  it  go  from  thence,  and  after  one  oscillation  suppose  it  to  return  to 
the  point  V  :  then  RV  will  be  the  retardation  arising  from  the  resistance 
of  the  air.  Of  this  RV  let  ST  be  a  fourth  part,  situated  in  the  middle. 
to  wit,  so  as  RS  and  TV  may  be  equal,  and  RS  may  be  to  ST  as  3  to  2 
then  will  ST  represent  very  nearly  the  retardation  during  the  descent 
from  S  to  A.  Restore  the  body  B  to  its  place:  and,  supjx  sing  the  body 
A  to  be  let  fall  from  the  point  S,  the  velocity  thereof  in  the  place  of  re 
flexion  A,  without  sensible  error,  will  be  the  same  as  if  it  had  descended 
m  vacit.o  from  the  point  T.  Upon  which  account  this  velocity  may  be 
represented  by  the  chord  of  the  arc  TA.  For  it  is  a  proposition  well 
known  to  geometers,  that  the  velocity  of  a  pendulous  body  in  the  loAvest 
point  is  as  the  chord  of  the  arc  which  it  has  described  in  its  descent.  Aftci 


OF    NATUltAL    PHILOSOPHY.  9  I 

reflexion,  suppose  the  body  A  comes  to  the  place  s,  and  the  body  B  to  the 
place  k.  Withdraw  the  body  B,  and  find  the  place  v,  from  which  if  the 
body  A,  being  let  go,  should  after  one  oscillation  return  to  the  place  r,  st 
may  be  a  fourth  part  of  rv.  so  placed  in  the  middle  thereof  as  to  leave  is 
equal  to  tv,  and  let  the  chord  of  the  arc  tA  represent  the  velocity  which 
the  body  A  had  in  the  place  A  immediately  after  reflexion.  For  t  will  be 
the  true  and  correct  place  to  which  the  body  A  should  have  ascended,  if 
the  resistance  of  the  air  had  been  taken  off.  In  the  s.ime  way  we  are  to 
correct  the  place  k  to  which  the  body  B  ascends,  by  finding  the  place  I  to 
which  it  should  have  ascended  in  vacuo.  And  thus  everything  may  be 
subjected  to  experiment,  in  the  same  manner  as  if  we  were  really  placed 
in  vacuo.  These  things  being  done,  we  are  to  take  the  product  (if  I  may 
so  say)  of  the  body  A,  by  the  chord  of  the  arc  TA  (which  represents  its 
velocity),  that  we  may  have  its  motion  in  the  place  A  immediately  before 
reflexion ;  and  then  by  the  chord  of  the  arc  /A,  that  we  may  have  its  mo 
tion  in  the  place  A  immediately  after  reflexion.  And  so  we  are  to  take 
the  product  of  the  body  B  by  the  chord  of  the  arc  B/,  that  we  may  have 
the  motion  of  the  same  immediately  after  reflexion.  And  in  like  manner, 
when  two  bodies  are  let  go  together  from  different  places,  we  are  to  find 
the  motion  of  each,  as  well  before  as  after  reflexion;  and  then  we  may 
compare  the  motions  between  themselves,  and  collect  the  effects  of  the  re 
flexion.  Thus  trying  the  thing  with  pendulums  of  ten  feet,  in  unequal 
as  well  as  equal  bodies,  and  making  the  bodies  to  concur  after  a  descent 
through  large  spaces,  as  of  8,  12,  or  16  feet,  I  found  always,  without  an 
error  of  3  inches,  that  when  the  bodies  concurred  together  directly,  equal 
changes  towards  the  contrary  parts  were  produced  in  their  motions,  and, 
of  consequence,  that  the  action  and  reaction  were  always  equal.  As  if  the 
body  A  impinged  upon  the  body  B  at  rest  with  9  parts  of  motion,  and 
losing  7,  proceeded  after  reflexion  with  2,  the  body  B  was  carried  back 
wards  with  those  7  parts.  If  the  bodies  concurred  with  contrary  motions, 
A  with  twelve  parts  of  motion,  and  B  with  six,  then  if  A  receded  with  J4, 
B  receded  with  8 ;  to  wit,  with  a  deduction  of  14  parts  of  motion  on 
each  side.  For  from  the  motion  of  A  subducting  twelve  parts,  nothing 
will  remain ;  but  subducting  2  parts  more,  a  motion  will  be  generated  of 
2  parts  towards  the  contrary  way ;  and  so,  from  the  motion  of  the  body 
B  of  6  parts,  subducting  14  parts,  a  motion  is  generated  of  8  parts  towards 
the  contrary  way.  But  if  the  bodies  were  made  both  to  move  towards  the 
same  way,  A,  the  swifter,  with  14  parts  of  motion,  B,  the  slower,  with  5, 
and  after  reflexion  A  went  on  with  5,  B  likewise  went  on  with  14  parts ; 
9  parts  being  transferred  from  A  to  B.  And  so  in  other  cases.  By  the 
congress  and  collision  of  bodies,  the  quantity  of  motion,  collected  from  the 
sum  of  the  motions  directed  towards  the  same  way,  or  from  the  difference, 
of  those  that  were  directed  towards  contrary  ways,  was  never  changed. 
For  the  error  of  an  inch  or  two  in  measures  may  be  easily  ascribed  to  tht 


92  THE    MATHEMATICAL    PRINCIPLES 

difficulty  of  executing  everything  with  accuracy.  It  was  not  easy  to  let 
go  the  two  pendulums  so  exactly  together  that  the  bodies  should  impinge 
one  upon  the  other  in  the  lowermost  place  AB ;  nor  to  mark  the  places  s, 
and  ky  to  which  the  bodies  ascended  after  congress.  Nay,  and  some  errors, 
too,  might  have  happened  from  the  unequal  density  of  the  parts  of  the  pen 
dulous  bodies  themselves,  and  from  the  irregularity  of  the  texture  pro 
ceeding  from  other  causes. 

But  to  prevent  an  objection  that  may  perhaps  be  alledged  against  the 
rule,  for  the  proof  of  which  this  experiment  was  made,  as  if  this  rule  did 
suppose  that  the  bodies  were  either  absolutely  hard,  or  at  least  perfectly 
elastic  (whereas  no  such  bodies  are  to  be  found  in  nature),  1  must  add.  that 
the  experiments  we  have  been  describing,  by  no  means  depending  upon 
that  quality  of  hardness,  do  succeed  as  well  in  soft  as  in  hard  bodies.  For 
if  the  rule  is  to  be  tried  in  bodies  not  perfectly  hard,  we  are  only  to  di 
minish  the  reflexion  in  such  a  certain  proportion  as  the  quantity  of  the 
elastic  force  requires.  By  the  theory  of  Wren  and  Huygens,  bodies  abso 
lutely  hard  return  one  from  another  with  the  same  velocity  with  which 
they  meet.  But  this  may  be  affirmed  with  more  certainty  of  bodies  per 
fectly  elastic.  In  bodies  imperfectly  elastic  the  velocity  of  the  return  is  to 
be  diminished  together  with  the  elastic  force ;  because  that  force  (except 
when  the  parts  of  bodies  are  bruised  by  their  congress,  or  suffer  some  such 
extension  as  happens  under  the  strokes  of  a  hammer)  is  (as  far  as  I  can  per 
ceive)  certain  and  determined,  and  makes  the  bodies  to  return  one  from 
the  other  with  a  relative  velocity,  which  is  in  a  given  ratio  to  that  relative 
velocity  with  which  they  met.  This  I  tried  in  balls  of  wool,  made  up 
tightly,  and  strongly  compressed.  For,  first,  by  letting  go  the  pendulous 
bodies,  and  measuring  their  reflexion,  I  determined  the  quantity  of  their 
elastic  force ;  and  then,  according  to  this  force,  estimated  the  reflexions 
that  ought  to  happen  in  other  cases  of  congress.  And  with  this  computa 
tion  other  experiments  made  afterwards  did  accordingly  agree ;  the  balls 
always  receding  one  from  the  other  with  a  relative  velocity,  which  was  to 
the  relative  velocity  with  which  they  met  as  about  5  to  9.  Balls  of  steel 
returned  with  almost  the  same  velocity  :  those  of  cork  with  a  velocity  some-^ 
thing  less ;  but  in  balls  of  glass  the  proportion  was  as  about  15  to  16. 
And  thus  the  third  Law,  so  far  as  it  regards  percussions  and  reflexions,  is 
proved  by  a  theory  exactly  agreeing  with  experience. 

In  attractions,  I  briefly  demonstrate  the  thing  after  this  manner.  Sup 
pose  an  obstacle  is  interposed  to  hinder  the  congress  of  any  two  bodies  A. 
B,  mutually  attracting  one  the  other :  then  if  either  body,  as  A,  is  more 
attracted  towards  the  other  body  B,  than  that  other  body  B  is  towards  the 
first  body  A,  the  obstacle  will  be  more  strongly  urged  by  the  pressure  of 
the  body  A  than  by  the  pressure  of  the  body  B,  and  therefore  will  not 
remain  in  equilibrio  :  but  the  stronger  pressure  will  prevail,  and  will  make 
the  system  of  the  two  bodies,  together  with  the  obstacle,  to  move  directly 


OF    NATURAL    PHILOSOPHY. 


93 


towards  the  parts  on  which  B  lies  ;  arid  in  free  spaces,  to  go  forward  in 
infmitiim  with  a  motion  perpetually  accelerated ;  which  is  absurd  and 
contrary  to  the  first  Law.  For,  by  the  first  Law,  the  system  ought  to  per 
severe  in  its  state  of  rest,  or  of  moving  uniformly  forward  in  a  right  line : 
and  therefore  the  bodies  must  equally  press  the  obstacle,  and  be  equally 
attracted  one  by  the  other.  I  made  the  experiment  on  the  loadstone  and 
iron.  If  these,  placed  apart  in  proper  vessels,  are  made  to  float  by  one 
another  in  standing  water,  neither  of  them  will  propel  the  other ;  but, 
by  being  equally  attracted,  they  will  sustain  each  other's  pressure,  and  rest 
at  last  in  an  equilibrium. 

So  the  gravitation  betwixt  the  earth  and  its  parts  is  mutual.  Let  the 
earth  FI  be  cut  by  any  plane  EG  into  two  parts  EGF 
and  EGI,  and  their  weights  one  towards  the  other 
will  be  mutually  equal.  For  if  by  another  plane 
HK,  parallel  to  the  former  EG,  the  greater  partFJ 
EGI  is  cut  into  two  parts  EGKH  and  HKI. 
whereof  HKI  is  equal  to  the  part  EFG,  first  cut 
oft',  it  is  evident  that  the  middle  part  EGKH,  will 
have  no  propension  by  its  proper  weight  towards  either  side,  but  will  hang 
as  it  were,  and  rest  in  an  equilibrium  betwixt  both.  But  the  one  extreme 
part  HKI  will  with  its  whole  weight  bear  upon  and  press  the  middle  part 
towards  the  other  extreme  part  EGF :  and  therefore  the  force  with  which 
EGI,  the  sum  of  the  parts  HKI  and  EGKH,  tends  towards  the  third  part 
EGF,  is  equal  to  the  weight  of  the  part  HKI,  that  is,  to  the  weight  of 
the  third  part  EGF.  And  therefore  the  weights  of  the  two  parts  EGI 
and  EGF,  one  towards  the  other,  are  equal,  as  I  was  to  prove.  And  in 
deed  if  those  weights  were  not  equal,  the  whole  earth  floating  in  the  non- 
resisting  aether  would  give  way  to  the  greater  weight,  and,  retiring  from 
it,  would  be  carried  off  in  infinitum. 

And  as  those  bodies  are  equipollent  in  the  congress  and  reflexion,  whose 
velocities  are  reciprocally  as  their  innate  forces,  so  in  the  use  of  mechanic 
instruments  those  agents  are  equipollent,  and  mutually  sustain  each  the 
contrary  pressure  of  the  other,  whose  velocities,  estimated  according  to  the 
determination  of  the  forces,  are  reciprocally  as  the  forces. 

So  those  weights  are  of  equal  force  to  move  the  arms  of  a  balance; 
which  during  the  play  of  the  balance  are  reciprocally  as  their  velocities 
upw  ards  and  downwards ;  that  is,  if  the  ascent  or  descent  is  direct,  those 
weights  are  of  equal  force,  which  are  reciprocally  as  the  distances  of  the 
points  at  which  they  are  suspended  from  the  axis  oi  the  balance  :  but  if 
they  are  turned  aside  by  the  interposition  of  oblique  planes,  or  other  ob 
stacles,  and  made  to  ascend  or  descend  obliquely,  those  bodies  will  be 
equipollent,  wThich  are  reciprocally  as  the  heights  of  their  ascent  and  de 
scent  taken  according  to  the  perpendicular ;  and  that  on  account  of  the 
determination  of  gravity  downwards. 


94  THE   MATHEMATICAL    PRINCIPLES 

And  in  like  manner  in  the  pully,  or  in  a  combination  of  pullies,  the 
force  of  a  hand  drawing  the  rope  directly,  which  is  to  the  weight,  whethel 
ascending  directly  or  obliquely,  as  the  velocity  of  the  perpendicular  ascent 
of  the  weight  to  the  velocity  of  the  hand  that  draws  the  rope,  will  sustain 
the  weight. 

In  clocks  and  such  like  instruments,  made  up  from  a  combination  of 
wheels,  the  contrary  forces  that  promote  and  impede  the  motion  of  the 
wheels,  if  they  are  reciprocally  as  the  velocities  of  the  parts  of  the  wheel 
on  which  they  are  impressed,  will  mutually  sustain  the  one  the  other. 

The  force  of  the  screw  to  press  a  body  is  to  the  force  of  the  hand  that 
turns  the  handles  by  which  it  is  moved  as  the  circular  velocity  of  the 
handle  in  that  part  where  it  is  impelled  by  the  hand  is  to  the  progressive 
velocity  of  the  screw  towards  the  pressed  body. 

The  forces  by  which  the  wedge  presses  or  drives  the  two  parts  of  the 
wood  it  cleaves  are  to  the  force  of  the  mallet  upon  the  wedge  as  the  pro- 
press  of  the  wedge  in  the  direction  of  the  force  impressed  upon  it  by  the 
mallet  is  to  the  velocity  with  which  the  parts  of  the  wood  yield  to  the 
wedge,  in  the  direction  of  lines  perpendicular  to  the  sides  of  the  wedge. 
And  the  like  account  is  to  be  given  of  all  machines. 

The  power  and  use  of  machines  consist  only  in  this,  that  by  diminishing 
the  velocity  we  may  augment  the  force,  and  the  contrary  :  from  whence 
in  all  sorts  of  proper  machines,  we  have  the  solution  of  this  problem ;  7 
move  a  given  weight  with  a  given  power,  or  with  a  given  force  to  over 
come  any  other  given  resistance.  For  if  machines  are  so  contrived  that  the 
velocities  of  the  agent  and  resistant  are  reciprocally  as  their  forces,  the 
agent  will  just  sustain  the  resistant,  but  with  a  greater  disparity  of  ve 
locity  will  overcome  it.  So  that  if  the  disparity  of  velocities  is  so  great 
as  to  overcome  all  that  resistance  which  commonly  arises  either  from  the 
attrition  of  contiguous  bodies  as  they  slide  by  one  another,  or  from  the 
cohesion  of  continuous  bodies  that  are  to  be  separated,  or  from  the  weights 
of  bodies  to  be  raised,  the  excess  of  the  force  remaining,  after  all  those  re 
sistances  are  overcome,  will  produce  an  acceleration  of  motion  proportional 
thereto,  as  well  in  the  parts  of  {he  machine  as  in  the  resisting  body.  But 
to  treat  of  mechanics  is  not  my  present  business.  I  was  only  willing  to 
show  by  those  examples  the  great  extent  and  certainty  of  the  third  Law  ot 
motion.  For  if  we  estimate  the  action  of  the  agent  from  its  force  and 
velocity  conjunctly,  and  likewise  the  reaction  of  the  impediment  conjuncth 
from  the  velocities  of  its  several  parts,  and  from  the  forces  of  resistance 
arising  from  the  attrition,  cohesion,  weight,  and  acceleration  of  those  parts, 
the  action  and  reaction  'YL  the  use  of  all  sorts  of  machines  will  b"  found 
always  equal  to  one  another.  And  so  far  as  the  action  is  propagated  by 
the  intervening  instruments,  and  at  last  impressed  upon  tic  resisting 
body,  the  ultimate  determination  of  the  action  will  be  always  contrary  to 
the  determination  of  the  reaction. 


OF    NATURAL    PHILOSOPHY 


95 


BOOK  I. 


OF  THE  MOTION  OF  BODIES. 


SECTION  I. 

Of  the  method  of  first  and  last  ratios  of  quantities,  by  the  help  wJicreoj 
we  demonstrate  the  propositions  that  follow. 

LEMMA    I. 

Quantities,  and  the  ratios  of  quantities,  which  in  any  finite  time  converge 
continually  to  equality,  and  before  the  end  of  that  time  approach  nearer 
the  one  to  the  other  than  by  any  given  difference,  become  ultimately 
equal. 

If  you  deny  it,  suppose  them  to  be  ultimately  unequal,  and  let  D  be 
their  ultimate  difference.  Therefore  they  cannot  approach  nearer  to 
equality  than  by  that  given  difference  D  ;  which  is  against  the  supposition, 

LEMMA  II. 

If  in  any  figure  AacE,  terminated  by  the  right  (f 
lines  A  a.  AE,  and  the  curve  acE,  there  be  in 
scribed  any  number  of  parallelograms  Ab,  Be, 
Cd,  fyc.,  comprehended  under  equal  bases  AB, 
BC,  CD,  ^«c.,  and  the  sides,  Bb,  Cc,  Dd,  ^c., 
parallel  to  one  side  Aa  of  the  figure ;  and  the 
parallelograms  aKbl,  bLcm,  cMdn,  §*c.,  are  com 
pleted.  Then  if  the  breadth  of  those  parallelo-  \ 
grams  be  supposed  to  be  diminisJied,  and  their  X  BF  C  D  |; 
number  to  be  augmented  in  infinitum :  /  say,  that  :he  ultimate  ratios 
'which  the  inscribed  fignre  AKbLcMdD,  the  tin  nmscribed  figure 
AalbmcndoE,  and  en  rvilijiear  figure  AabcdE,  will  have  to  one  another, 
are  ratios  of  equality. 

For  the  difference  of  the  inscribed  and  circumscribed  figures  is  the  sum 
of  the  parallelograms  K7,  Lw,  M//.  Do.  that  is  (from  the  equality  of  all 
their  bases),  the  rectangle  under  one  of  their  bases  K6  and  the  sum  of  their 
altitudes  Aa,  that  is,  the  rectangle  ABla.  But  this  rectangle,  because 


M 


a 


96 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK    1 


its  breadth  AB  is  supposed  diminished  •  in  infinitum,  becomes  less  than 
any  given  space.  And  therefore  (by  Lem.  I)  the  figures  inscribed  and 
circumscribed  become  ultimately  equal  one  to  the  other;  and  much  more 
will  the  intermediate  curvilinear  figure  be  ultimately  equal  to  either* 
Q.E.D. 

LEMMA  III. 

The  same  ultimate  ratios  are  also  ratios  of  equality,  when  the  breadth^ 
AB,  BC,  DC,  fyc.,  of  the  parallelograms  are  unequal,  and  are  all  di 
minished  in  infinitum. 

For  suppose  AF  equal  to  the  greatest  breadth,  and 
complete  the  parallelogram  FAaf.  This  parallelo 
gram  will  be  greater  than  the  difference  of  the  in 
scribed  and  circumscribed  figures ;  but,  because  its 
breadth  AF  is  diminished  in  infinitum,  it  will  be 
come  less  than  any  given  rectangle.  Q.E.D. 

COR.  1.  Hence  the  ultimate  sum  of  those  evanes 
cent  parallelograms  will  in  all  parts  coincide  with 

the  curvilinear  figure.  A     BF     C      D      E 

COR.  2.  Much  more  will  the  rectilinear  figure^comprehendcd  under  tne 
chords  of  the  evanescent  arcs  ab,  be,  cd,  (fee.,  ultimately  coincide  with  tl.c 
curvilinear  figure. 

COR.  3.  And  also  the  circumscribed  rectilinear  figure  comprehended 
under  the  tangents  of  the  same  arcs. 

COR.  4  And  therefore  these  ultimate  figures  (as  to  their  perimeters  acE) 
are  not  rectilinear,  but  curvilinear  limi's  of  rectilinear  figures. 

LEMMA  IV. 

If  in  two  figures  AacE,  PprT,  you  inscribe  (as  before) 

two  ranks  of  parallelograms,  an  equal  number  in 

each  rank,  and,  when  their  breadths  are  diminished 

in  infinitum.  the  ultimate  ratios  of  the  parallelograms 

in  one  figure  to  those  in  the  other,  each  to  each  respec 
tively,  are  the  same;  I  say,  that  those  two  figures 

AacE,  PprT,  are  to  one  another  in  that  same  ratio. 

For  as  the  parallelograms  in  the  one  are  severally  to  p 
the  parallelograms  in  the  other,  so  (by  composition)  is  the  < 
sum  of  all  in  the  one  to  the  sum  of  all  in  the  other  :  and 
so  is  the  one  figure  to  the  other;  because  (by  Lem.  Ill)  the 
former  figure  to  the  former  sum,  and  the  latter  figure  to  the 
latter  sum,  are  both  in  the  ratio  of  equality.    Q.E.D. 

COR.  Hence  if  two  quantities  of  any  kind  are  any 
how  divided  into  an  equal  number  of  parts,  and  those  A 


£ 


SEC.     I.]  OF    NATURAL    PHILOSOPHY.  97 

parts,  when  their  number  is  augmented,  and  their  magnitude  diminished 
in  infinitum,  have  a  given  ratio  one  to  the  other,  the  first  to  the  first,  the 
second  to  the  second,  and  so  on  in  order,  the  whole  quantities  will  be  one  to 
the  other  in  that  same  given  ratio.  For  if,  in  the  figures  of  this  Lemma, 
the  parallelograms  are  taken  one  to  the  other  in  the  ratio  of  the  parts,  the 
sum  of  the  parts  will  always  be  as  the  sum  of  the  parallelograms ;  and 
therefore  supposing  the  number  of  the  parallelograms  and  parts  to  be  aug 
mented,  and  their  magnitudes  diminished  in  infinitum,  those  sums  will  be 
in  the  ultimate  ratio  of  the  parallelogram  in  the  one  figure  to  the  corres 
pondent  parallelogram  in  the  other  ;  that  is  (by  the  supposition),  in  the 
ultimate  ratio  of  any  part  of  the  one  quantity  to  the  correspondent  part  of 
the  other. 

LEMMA  V. 

In  similar  figures,  all  sorts  of  homologous  sides,  whether  curvilinear  or 
rectilinear,  are  proportional ;  and  the  areas  are  in  the  duplicate  ratio 
of  the  homologous  sides. 

LEMMA  VI. 

If  any  arc  ACB,  given  in  position,  is  snb-  _j 
tended  by  its  chord  AB,  and  in  any  point 
A,  in  the  middle  of  the  contiinied  curva 
ture,  is  touched  by  a  right  line  AD,  pro 
duced  both  ways ;  then  if  the  points  A    R 
and  B  approach  one  another  and  meet, 
I  say,  the  angle  RAT),  contained  between, 
the  chord  and  the  tangent,  will  be  dimin-     ? 
ished  in  infinitum,  a/id  ultimately  will  vanish. 

For  if  that  angle  does  not  vanish,  the  arc  ACB  will  contain  with  the 
tangent  AD  an  angle  equal  to  a  rectilinear  angle ;  and  therefore  the  cur 
vature  at  the  point  A  will  not  be  continued,  which  is  against  the  supposi 
tion. 

LEMMA  VII. 

The  same  things  being  supposed,  I  say  that  the  ultimate  ratio  of  the  arc, 

chord,  and  tangent,  any  one  to  any  other,  is  the  ratio  of  equality. 

For  while  the  point  B  approaches  towards  the  point  A,  consider  always 

AB  and  AD  as  produced  to  the  remote  points  b  and  d,  and  parallel  to  the 

secant  BD  draw  bd :  and  let  the  arc  Acb  be  always  similar  to   the  arc 

ACB.     Then,  supposing  the  points  A  and  B  to  coincide,  the  angle  dAb 

will  vanish,  by  the  preceding  Lemma;  and  therefore  the  right  lines  Ab, 

Arf  (which  are  always  finite),  and  the  intermediate  arc  Acb,  will  coincide, 

and  become  equal  among  themselves.     Wheref  ,re,  the  right  lines  AB,  AD, 


98  THE    MATHEMATICAL    PRINCIPLES  [SEC.    I. 

and  the  intermediate  arc  ACB   (which  are  always  proportional  to  the 
former),  will  vanish,  and  ultimately  acquire  the  ratio  of  equality.    Q.E.D. 

COR.  1.  Whence  if  through  B  we  draw  A 

BP  parallel  to  the  tangent,  always  cutting 
any  right  line  AF  passing  through  A  in  F/— i- 

P,  this  line  BP  will  be  ultimately  in  the 

ratio  of  equality  with  the  evanescent  arc  ACB ;  because,  completing  the 
parallelogram  APBD,  it  is  always  in  a  ratio  of  equality  with  AD. 

COR.  2.  And  if  through  B  and  A  more  right  lines  are  drawn,  as  BE, 
I5D,  AF,  AG,  cutting  the  tangent  AD  and  its  parallel  BP  :  the  ultimate 
ratio  of  all  the  abscissas  AD,  AE,  BF,  BG,  and  of  the  chord  and  arc  AB, 
any  one  to  any  other,  will  be  the  ratio  of  equality. 

COR.  3.  And  therefore  in  all  our  reasoning  about  ultimate  ratios,  we 
may  freely  use  any  one  of  those  lines  for  any  other. 

LEMMA  VIII. 

If  the  right  lines  AR,  BR,  with  the  arc  ACB,  the  chord  AB,  and  the 
tangent  AD,  constitute  three  triangles  RAB.  RACB,  RAD,  and  the 
points  A  and  B  approach  and  meet :  I  say,  that  the  ultimate  form  oj 
these  evanescent  triangles  is  that  of  similitude,  and  their  ultimate 
ratio  that  of  equality. 

For  while  the  point  B  approaches  towards  A 
the  point  A,  consider  always  AB,  AD,  AR, 
as  produced  to  the  remote  points  b,  d,  and  r, 
and  rbd  as  drawn  parallel  to  RD,  and  let 
the  arc  Acb  be  always  similar  to  the  arc 
ACB.  Then  supposing  the  points  A  and  B 
to  coincide,  the  angle  bAd  will  vanish ;  and 
therefore  the  three  triangles  rAb,  rAcb,rAd 
^which  are  always  finite),  will  coincide,  and  on  that  account  become  both 
similar  and  equal.  And  therefore  the  triangles  RAB.  RACB,  RAD 
which  are  always  similar  and  proportional  to  these,  will  ultimately  be 
come  both  similar  and  equal  among  themselves.  Q..E.D. 

COR.  And  hence  in  all  reasonings  about  ultimate  ratios,  we  may  indif 
ferently  use  any  one  of  those  triangles  for  any  other. 

LEMMA  IX. 

If  a  ngnt  line  AE.  and  a  curve  tine  ABC,  both  given  by  position,  cut 
each  other  in  a  given  angle,  A ;  and  to  that  right  line,  in  another 
given  angle,  BD,  CE  are  ordinately  applied,  meeting  the  curve  in  B, 
C :  and  the  points  B  and  C  together  approach  towards  and  meet  in 
the  point  A :  /  say,  that  the  areas  of  the  triangles  ABD,  ACE,  wilt 
ultimately  be  one  to  the  other  in  the  duplicate  ratio  of  the  sides. 


BOOK  LI 


OF    NATURAL    PHILOSOPHY. 


For  while  the  points  B,  C,  approach 
towards  the  point  A,  suppose  always  AD 
to  be  produced  to  the  remote  points  d  and  . 
e,  so  as  Ad,  Ae  may  be  proportional  to 
AD,  AE ;  and  the  ordinates  db,  ec,  to  be 
drawn  parallel  to  the  ordinates  DB  and 
EC,  and  meeting  AB  and  AC  produced  D 
in  b  and  c.  Let  the  curve  A  be  be  similar 
to  the  curve  A  BC,  and  draw  the  right  line 
Ag-  so  as  to  touch  both  curves  in  A,  and 
cut  the  ordinates  DB,  EC,  db  ec,  in  F,  G, 
J]  g.  Then,  supposing  the  length  Ae  to  remain  the  same,  let  the  points  B 
and  C  meet  in  the  point  A ;  and  the  angle  cAg  vanishing,  the  curvilinear 
areas  AW,  Ace  will  coincide  with  the  rectilinear  areas  A/rf,  Age ;  and 
therefore  (by  Lem.  V)  will  be  one  to  the  other  in  the  duplicate  ratio  of 
the  sides  Ad,  Ae.  But  the  areas  ABD,  ACE  are  always  proportional  to 
these  areas ;  and  so  the  sides  AD,  AE  are  to  these  sides.  And  therefore 
the  areas  ABD,  ACE  are  ultimately  one  to  the  other  in  the  duplicate  ratio 
of  the  sides  AD,  AE.  Q.E.D. 

LEMMA  X. 

The  spaces  which  a  bodij  describes  by  any  finite  force  urging  it.  whether 

that  force  is  determined  and  immutable,  or  is  continually  augmented 

or  continually  diminished,  are  in  the  very  beginning  of  the  motion  one 

to  the  other  in  the  duplicate  ratio  of  the  times. 

Let  the  times  be  represented  by  the  lines  AD,  AE,  and  the  velocities 
generated  in  those  times  by  the  ordinates  DB,  EC.  The  spaces  described 
with  these  velocities  will  be  as  the  areas  ABD,  ACE.  described  by  those 
ordinates,  that  is,  at  the  very  beginning  of  the  motion  (by  Lem.  IX),  in 
the  duplicate  ratio  of  the  times  AD,  AE.  Q..E.D. 

COR.  1.  And  hence  one  may  easily  infer,  that  the  errors  of  bodies  des 
cribing  similar  parts  of  similar  figures  in  proportional  times,  are  nearly 
as  the  squares  of  the  times  in  which  they  are  generated  ;  if  so  be  these 
errors  are  generated  by  any  equal  forces  similarly  applied  to  the  bodies, 
and  measured  by  the  distances  of  the  bodies  from  those  places  of  the  sim 
ilar  figures,  at  which,  without  the  action  of  those  forces,  the  bodies  would 
have  arrived  in  those  proportional  times. 

COR.  2.  But  the  errors  that  are  generated  by  proportional  forces,  sim 
ilarly  applied  to  the  bodies  at  similar  parts  of  the  similar  figures,  are  as 
the  forces  and  the  squares  of  the  times  conjuiu  tly. 

COR.  3.  The  same  thing  is  to  be  understood  of  any  spaces  whatsoever 
described  by  bodies  urged  with  different  forces ;  all  which,  in  the  very  be- 
g'nning  of  the  motion,  are  as  the  forces  and  the  squares  of  the  times  conjunctly. 


100 


THE    MATHEMATICAL    PRINCIPLES 


I  SEC.  1 


COR.  4.  And  therefore  the  forces  are  as  the  spaces  described  in  the  very 
beginning  of  the  motion  directly,  and  the  squares  of  the  times  inversely. 

COR.  5.  And  the  squares  of  the  times  are  as  the  spaces  described  direct 
ly,  und  the  forces  inversely. 

SCHOLIUM. 

If  in  comparing  indetermined  quantities  of  different  sorts  one  with 
another,  any  one  is  said  to  be  as  any  other  directly  or  inversely,  the  mean 
ing  is,  that  the  former  is  augmented  or  diminished  in  the  same  ratio  with 
the  latter,  or  with  its  reciprocal.  And  if  any  one  is  said  to  be  as  any  other 
two  or  more  directly  or  inversely,  the  meaning  is,  that  the  first  is  aug 
mented  or  diminished  in  the  ratio  compounded  of  the  ratios  in  which  the 
others,  or  the  reciprocals  of  the  others,  are  augmented  or  diminished.  As 
if  A  is  said  to  be  as  B  directly,  and  C  directly,  and  D  inversely,  the  mean 
ing  is,  that  A  is  augmented  or  diminished  in  the  same  ratio  with  B  X  C 
X  -jj-,  that  is  to  say,  that  A  and  -£•  arc  one  to  the  other  in  a  given  ratio. 

LEMMA  XL 

The  evanescent  subtense  of  the  angle  of  contact,  in  all  curves  which  at 
the  point  of  contact  have  a  finite  curvature,  is  ultimately  in  the  dupli 
cate  rati1)  of  the  subtense  of  the  conterminate  arc. 
CASE  1.  Let  AB  be  that  arc,  AD  its  tangent,  BD 
the  subtense  of  the  angle  of  contact  perpendicular  on 
the  tangent,  AB  the  subtense  of  the  arc.  Draw  BG 
perpendicular  to  the  subtense  AB,  and  AG  to  the  tan 
gent  AD,  meeting  in  G  ;  then  let  the  points  D,  B,  and 
G.  approach  to  the  points  d,  b,  and  g,  and  suppose  J 
to  be  the  ultimate  intersection  of  the  lines  BG,  AG, 
when  the  points  D,  B,  have  come  to  A.  It  is  evident 
that  the  distance  GJ  may  be  less  than  any  assignable. 
But  (from  the  nature  of  the  circles  passing  through 
the  points  A,  B,  G,  A,  b,  g,)  AE2=  AG  X  BD,  and 
A62  =  Ag  X  bd ;  and  therefore  the  ratio  of  AB2  to  Ab2  is  compounded  oi 
the  ratios  of  AG  to  Ag,  and  of  Ed  to  bd.  But  because  GJ  may  be  as 
sumed  of  less  length  than  any  assignable,  the  ratio  of  AG  to  Ag  may  be 
such  as  to  differ  from  the  ratio  of  equality  by  less  than  any  assignable 
difference ;  and  therefore  the  ratio  of  AB2  to  Ab2  may  be  such  as  to  differ 
from  the  ratio  of  BD  to  bd  by  less  than  any  assignable  difference.  There 
fore,  by  Lem.  I,  the  ultimate  ratio  of  AB2  to  Ab2  is  the  same  with  tho  ul 
timate  ratio  of  BD  to  bd.  Q.E.D. 

CASE  2.  Now  let  BD  be  inclined  to  AD  in  any  given  an*r1r,  and  the 
ultimate  ratio  of  BD  to  bd  will  always  be  the  same  as  before,  and  there 
fore  the  same  with  the  ratio  of  AB2  to  Ab2.  Q.E-P 


BOOK  I.] 


OF    NATURAL    PHILOSOPHY. 


101 


CASE  3.  And  if  we  suppose  the  angle  D  not  to  be  given,  but  that  the 
right  line  BD  converges  to  a  given  point,  or  is  determined  by  any  other 
condition  whatever  ;  nevertheless  the  angles  D,  d,  being  determined  by  the 
same  law,  will  always  draw  nearer  to  equality,  arid  approach  nearer  to 
each  other  than  by  any  assigned  difference,  and  therefore,  by  Lem.  I,  will  at 
lust  be*  equal ;  and  therefore  the  lines  BD;  bd  arc  in  the  same  ratio  to  each 
other  as  before.  Q.E.D. 

COR.  1.  Therefore  since  the  tangents  AD,  Ad,  the  arcs  AB,  Ab,  and 
their  sines,  BC,  be,  become  ultimately  equal  to  the  chords  AB,  Ab}  their 
squares  will  ultimately  become  as  the  subtenses  BD,  bd. 

COR.  2.  Their  squares  are  also  ultimately  as  the  versed  sines  of  the  arcs, 
bisecting  the  chords,  and  converging  to  a  given  point.  For  those  versed 
sines  are  as  the  subtenses  BD,  bd. 

COR.  3.  And  therefore  the  versed  sine  is  in  the  duplicate  ratio  of  the 
time  in  which  a  body  will  describe  the  arc  with  a  given  velocity. 

COR.  4.  The  rectilinear  triangles  ADB,  Adb  are 
ultimately  in  the  triplicate  ratio  of  the  sides  AD,  Ad,  c 
and  in  a  sesquiplicate  ratio  of  the  sides  DB,  db  ;  as 
being  in  the  ratio  compounded  of  the  sides  AD  to  DB, 
and  of  Ad  to  db.  So  also  the  triangles  ABC,  Abe 
are  ultimately  in  the  triplicate  ratio  of  the  sides  BC,  be. 
What  I  call  the  sesquiplicate  ratio  is  the  subduplicate 
of  the  triplicate,  as  being  compounded  of  the  simple 
and  subduplicate  ratio.  j 

COR.  5.  And  because  DB,  db  are  ultimately  paral-  g 
lei  and  in  the  duplicate  ratio  of  the  lines  AD,  Ad,  the 
ultimate  curvilinear  areas  ADB,  Adb  will  be  (by  the  nature  of  the  para 
bola)  two  thirds  of  the  rectilinear  triangles  ADB,  Adb  and  the  segments 
AB,  Ab  will  be  one  third  of  the  same  triangles.  And  thence  those  areas 
and  those  segments  will  be  in  the  triplicite  ratio  as  well  of  the  tangents 
AD,  Ad,  as  of  the  chords  and  arcs  AB,  AB. 

SCHOLIUM. 

But  we  have  all  along  supposed  the  angle  of  contact  to  be  neither  infi 
nitely  greater  nor  infinitely  less  than  the  angles  of  contact  made  by  cir 
cles  and  their  tangents ;  that  is,  that  the  curvature  at  the  point  A  is  neither 
infinitely  small  nor  infinitely  great,  or  that  the  interval  AJ  is  of  a  finite  mag 
nitude.  For  DB  may  be  taken  as  AD3 :  in  which  case  no  circle  can  be  drawn 
through  the  point  A,  between  the  tangent  AD  and  the  curve  AB,  and 
therefore  the  angle  of  contact  will  be  infinitely  less  than  those  of  circles. 
And  by  a  like  reasoning,  if  DB  be  made  successfully  as  AD4,  AD5,  AD8, 
AD7,  etc.,  we  shall  have  a  series  of  angles  of  contact,  proceeding  in  itifini- 
tum,  wherein  every  succeeding  term  is  infinitely  less  than  the  pre- 


102  THE    MATHEMATICAL    PRINCIPLES  [BOOK  1 

ceding.  And  if  DB  be  made  successively  as  AD2,  AD|,  AD^,  AD],  AD| 
AD7,  &c.,  we  shall  have  another  infinite  series  of  angles  of  contact,  the  first 
of  which  is  of  the  same  sort  with  those  of  circles,  the  second  infinitely 
greater,  and  every  succeeding  one  infinitely  greater  than  the  preceding. 
But  between  any  two  of  these  angles  another  series  of  intermediate  angles 
of  contact  may  be  interposed,  proceeding  both  ways  in  infinitum.  wherein 
every  succeeding  angle  shall  be  infinitely  greater  or  infinitely  less  than  the 
preceding.  As  if  between  the  terms  AD2  and  AD3  there  were  interposed 
the  series  AD'f,  ADy,  AD49,  AD|,  AD?,  AD|,  AD^1,  AD^,  AD^7,  &c.  And 
again,  between  any  two  angles  of  this  series,  a  new  series  of  intermediate 
angles  may  be  interposed,  differing  from  one  another  by  infinite  intervals. 
Nor  is  nature  confined  to  any  bounds. 

Those  things  which  have  been  demonstrated  of  curve  lines,  and  the 
euperfices  which  they  comprehend,  may  be  easily  applied  to  the  curve  su- 
perfices  and  contents  of  solids.  These  Lemmas  are  premised  to  avoid  the 
tediousness  of  deducing  perplexed  demonstrations  ad  absurdnm,  according 
to  the  method  of  the  ancient  geometers.  For  demonstrations  are  more 
contracted  by  the  method  of  indivisibles :  but  because  the  hypothesis  of 
indivisibles  seems  somewhat  harsh,  and  therefore  that  method  is  reckoned 
less  geometrical,  I  chose  rather  to  reduce  the  demonstrations  of  the  follow 
ing  propositions  to  the  first  and  last  sums  and  ratios  of  nascent  and  evane 
scent  quantities,  that  is,  to  the  limits  of  those  sums  and  ratios  ;  and  so  to 
premise,  as  short  as  I  could,  the  demonstrations  of  those  limits.  For  hereby 
the  same  thing  is  performed  as  by  the  method  of  indivisibles ;  and  now 
those  principles  being  demonstrated,  we  may  use  them  with  more  safety. 
Therefore  if  hereafter  I  should  happen  to  consider  quantities  as  made  up  of 
particles,  or  should  use  little  curve  lines  for  right  ones,  I  would  not  be  un- 
(lerstood  to  mean  indivisibles,  but  evanescent  divisible  quantities  :  not  the 
sums  and  ratios  of  determinate  parts,  but  always  the  limits  of  sums  and 
ratios ;  and  that  the  force  of  such  demonstrations  always  depends  on  the 
method  laid  down  in  the  foregoing  Lemmas. 

Perhaps  it  may  be  objected,  that  there  is  no  ultimate  proportion,  of 
evanescent  quantities ;  because  the  proportion,  before  the  quantities  have 
vanished,  is  not  the  ultimate,  and  when  they  are  vanished,  is  none.  But 
by  the  same  argument,  it  may  be  alledged,  that  a  body  arriving  at  a  cer 
tain  place,  and  there  stopping  has  no  ultimate  velocity :  because  the  velo 
city,  before  the  body  comes  to  the  place,  is  not  its  ultimate  velocity  ;  when 
it  has  arrived,  is  none  i  ut  the  answer  is  easy;  for  by  the  ultimate  ve 
locity  is  meant  that  with  which  the  body  is  moved,  neither  before  it  arrives 
at  its  last  place  and  the  motion  ceases,  nor  after,  but  at  the  very  instant  it 
arrives  ;  that  is,  that  velocity  with  which  the  body  arrives  at  its  last  place, 
and  with  which  the  motion  ceases.  And  in  like  manner,  by  the  ultimate  ra 
tio  of  evanescent  quantities  is  to  Le  understood  the  ratio  of  the  ijuantitiea 


SEC.  II.]  OF    NATURAL    PHILOSOPHY.  103 

not  before  they  vanish,  nor  afterwards,  but  with  which  they  vanish.  In 
like  manner  the  first  ratio  of  nascent  quantities  is  that  with  which  they  begin 
to  be.  And  the  first  or  last  sum  is  that  with  which  they  begin  and  cease 
to  be  (or  to  be  augmented  or  diminished).  There  is  a  limit  which  the  ve 
locity  at  the  end  of  the  motion  may  attain,  but  not  exceed.  This  is  the 
ultimate  velocity.  And  there  is  the  like  limit  in  all  quantities  and  pro 
portions  that  begin  and  cease  to  be.  And  since  such  limits  are  certain  and 
definite,  to  determine  the  same  is  a  problem  strictly  geometrical.  But 
whatever  is  geometrical  we  may  be  allowed  to  use  in  determining  and  de 
monstrating  any  other  thing  that  is  likewise  geometrical. 

It  may  also  be  objected,  that  if  the  ultimate  ratios  of  evanescent  quan 
tities  are  given,  their  ultimate  magnitudes  will  be  also  given  :  and  so  all 
quantities  will  consist  of  indivisibles,  which  is  contrary  to  what  Euclid 
has  demonstrated  concerning  incommensurables,  in  the  10th  Book  of  his 
Elements.  But  this  objection  is  founded  on  a  false  supposition.  For 
those  ultimate  ratios  with  which  quantities  vanish  are  not  truly  the  ratios 
of  ultimate  quantities,  but  limits  towards  which  the  ratios  of  quantities 
decreasing  without  limit  do  always  converge  ;  and  to  which  they  approach 
nearer  than  by  any  given  difference,  but  never  go  beyond,  nor  in  effect  attain 
to,  till  the  quantities  are  diminished  in  wfinitum.  This  thing  will  appear 
more  evident  in  quantities  infinitely  great.  If  two  quantities,  whose  dif 
ference  is  given,  be  augmented  in  infin&um,  the  ultimate  ratio  of  these 
quantities  will  be  given,  to  wit,  the  ratio  of  equality  ;  but  it  does  not  from 
thence  follow,  that  the  ultimate  or  greatest  quantities  themselves,  whose 
ratio  that  is,  will  be  given.  Therefore  if  in  what  follows,  for  the  sake  of 
being  more  easily  understood,  I  should  happen  to  mention  quantities  as 
least,  or  evanescent,  or  ultimate,  you  are  not  to  suppose  that  quantities  of 
any  determinate  magnitude  are  meant,  but  such  as  are  conceived  to  be  al 
ways  diminished  without  end. 


SECTION  II. 

Of  the  Invention  of  Centripetal  Forces. 

PROPOSITION  I.    THEOREM  1. 

The  areas,  which  revolving  bodies  describe  by  radii  drawn  to  an  ^mmo- 

vable  centra  of  force  do  lie  in  tJ:e  same  immovable  planes,  and  are  pro- 

portional  to  the  times  in  which  they  are  described. 

For  suppose  the  time  to  be  divided  into  equal  parts,  and  in  the  first  part 

of  that  time  let  the  body  by  its  innate  force  describe  the  right  line  AB 

In  the  second  part  of  that  time,  the  same  would  (by  Law  I.),  if  not  hindered, 

proceel  directly  to  c,  alo  ILJ;  the  line  Be  equal  to  AB  ;  so  that  by  the  radii 

AS,  BS,  cS,  draw. i  to  the  centre,  the  equal  areas  ASB,  BSc,  would  be  de- 


104 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK    I 


scribed.  But  when  the  body 
is  arrived  at  B,  suppose 
that  a  centripetal  force  acts 
at  once  with  a  great  im 
pulse,  and,  turning  aside  the 
body  from  the  right  line  Be, 
compels  it  afterwards  to  con 
tinue  its  motion  along  the 
right  line  BC.  Draw  cC 
parallel  to  BS  meeting  BC 
in  C ;  and  at  the  end  of  the 
second  part  of  the  time,  the 
body  (by  Cor.  I.  of  the  Laws) 
will  be  found  in  C,  in  the 
same  plane  with  the  triangle 
A  SB.  Join  SC,  and,  because  s 
SB  and  Cc  are  parallel,  the  triangle  SBC  will  be  equal  to  the  triangle  SBc, 
and  therefore  also  to  the  triangle  SAB.  By  the  like  argument,  if  the 
centripetal  force  acts  successively  in  C,  D,  E.  &c.,  and  makes  the  body,  in 
each  single  particle  of  time,  to  describe  the  right  lines  CD,  DE,  EF7  &c., 
they  will  all  lie  in  the  same  plane :  and  the  triangle  SCD  will  be  equal  to 
the  triangle  SBC,  and  SDE  to  SCD,  and  SEF  to  SDE.  And  therefore, 
in  equal  times,  equal  areas  are  described  in  one  immovable  plane :  and,  by 
composition,  any  sums  SADS,  SAFS,  of  those  areas,  are  one  to  the  other 
as  the  times  in  which  they  are  described.  Now  let  the  number  of  those 
triangles  be  augmented,  and  their  breadth  diminished  in  wjinitum  ;  and 
(by  Cor.  4,  Lem.  III.)  their  ultimate  perimeter  ADF  will  be  a  curve  line : 
and  therefore  the  centripetal  force,  by  which  the  body  is  perpetually  drawn 
back  from  the  tangent  of  this  curve,  will  act  continually  ;  and  any  described 
areas  SADS,  SAFS,  which  are  always  proportional  to  the  times  of  de 
scription,  will,  in  this  case  also,  be  proportional  to  those  times.  Q.E.D. 

COR.  1.  The  velocity  of  a  body  attracted  towards  an  immovable  centre, 
in  spaces  void  of  resistance,  is  reciprocally  as  the  perpendicular  let  fall 
from  that  centre  on  the  right  line  that  touches  the  orbit.  For  the  veloci 
ties  in  those  places  A,  B,  C,  D,  E.  are  as  the  bases  AB,  BC,  CD,  DE,  EF. 
of  equal  triangles  ;  and  these  bases  are  reciprocally  as  the  perpendiculars 
let  fall  upon  them. 

COR.  2.  If  the  chords  AB,  BC  of  two  arcs,  successively  described  in 
equal  times  by  the  same  body,  in  spaces  void  of  resistance,  are  completed 
into  a  parallelogram  ABCV,  and  the  diagonal  BV  of  this  parallelogram; 
in  the  position  which  it  ultimately  acquires  when  those  arcs  are  diminished 
in  irifinitum,  is  produced  both  ways,  it  will  pass  through  the  centre  of  force. 

COR.  3.  If  the  chords  AB,  BC,  and  DE,  EF,  cf  arcs  described  in  equal 


SEC.  II.] 


OF    NATURAL    PHILOSOPHY. 


105 


times,  in  spaces  void  of  resistance,  are  completed  into  the  parallelograms 
ABCV,  DEFZ  :  the  forces  in  B  and  E  are  one  to  the  other  in  the  ulti 
mate  ratio  of  the  diagonals  BV,  EZ,  when  those  arcs  are  diminished  in 
infinitum.  For  the  motions  BC  and  EF  of  the  body  (by  Cor.  1  of  the 
Laws)  are  compounded  of  the  motions  Be,  BV,  and  E/",  EZ  :  but  BV  and 
EZ,  which  are  equal  to  Cc  and  F/,  in  the  demonstration  of  this  Proposi 
tion,  were  generated  by  the  impulses  of  the  centripetal  force  in  B  and  E; 
and  are  therefore  proportional  to  those  impulses. 

COR.  4.  The  forces  by  which  bodies,  in  spaces  void  of  resistance,  are 
drawn  back  from  rectilinear  motions,  and  turned  into  curvilinear  orbits, 
are  one  to  another  as  the  versed  sines  of  arcs  described  in  equal  times ;  which 
versed  sines  tend  to  the  centre  of  force,  and  bisect  the  chords  when  those 
arcs  are  diminished  to  infinity.  For  such  versed  sines  are  the  halves  of 
the  diagonals  mentioned  in  Cor.  3. 

COR.  5.  And  therefore  those  forces  are  to  the  force  of  gravity  as  the  said 
versed  sines  to  the  versed  sines  perpendicular  to  the  horizon  of  those  para 
bolic  arcs  which  projectiles  describe  in  the  same  time. 

COR.  6.  And  the  same  things  do  all  hold  good  (by  Cor.  5  of  the  Laws), 
when  the  planes  in  which  the  bodies  are  moved,  together  with  the  centres 
of  force  which  are  placed  in  those  planes,  are  not  at  rest,  but  move  uni 
formly  forward  in  right  lines. 

PROPOSITION  II.     THEOREM  II. 

Every  body  that  moves  in  any  curve  line  described  in  a  plane,  and  by  a 
radius,  drawn  to  a  point  either  immovable,  or  moving  forward  with 
an  uniform  rectilinear  motion,  describes  about  that  point  areas  propor 
tional  to  the  times,  is  urged  by  a  centripetal  force  directed  to  that  point 
CASE.  1.  For  every  body 

that  moves  in  a  curve  line, 

is  (by  Law  1)  turned  aside 

from  its   rectilinear   course 

by  the  action  of  some  force 

that  impels  it.  And  that  force 

by  which  the  body  is  turned 

off  from  its  rectilinear  course, 

and  is  made  to  describe,  in 

equal  times,  the  equal  least 

triangles  SAB,  SBC,  SCD, 

&c.,   about    the   immovable 

point  S  (by  Prop.  XL.  Book 

1,  Elem.  and  Law  II),   acts 

in  the  place  B,  according  to 


the  direction  of  a  line 


par- 


1U6  THE    MATHEMATICAL    PRINCIPLES  [BOOK    f. 

allel  K  cC.  that  is,  in  the  direction  of  the  line  BS.  and  in  the  place  C, 
accordii  g  to  the  direction  of  a  line  parallel  to  dD,  that  is,  in  the  direction 
of  the  line  CS,  (fee.;  and  therefore  acts  always  in  the  direction  of  lines 
tending  to  the  immovable  point  S.  Q.E.I). 

CASE.  2.  And  (by  Cor.  5  of  the  Laws)  it  is  indifferent  whether  the  su- 
perfices  in  which  a  body  describes  a  curvilinear  figure  be  quiescent,  or  moves 
together  with  the  body,  the  figure  described,  and  its  point  S,  uniformly 
forward  in  right  lines. 

COR.  1.  In  non-resisting  spaces  or  mediums,  if  the  areas  are  not  propor 
tional  to  the  times,  the  forces  are  not  directed  to  the  point  in  which  the 
radii  meet ;  but  deviate  therefrom  in.  consequently  or  towards  the  parts  to 
which  the  motion  is  directed,  if  the  description  of  the  areas  is  accelerated ; 
but  in  antecedentia,  if  retarded. 

COR.  2.  And  even  in  resisting  mediums,  if  the  description  of  the  areas 
is  accelerated,  the  directions  of  the  forces  deviate  from  the  point  in  which 
the  radii  meet,  towards  the  parts  to  which  the  motion  tends. 

SCHOLIUM. 

A  body  may  be  urged  by  a  centripetal  force  compounded  of  several 
forces ;  in  which  case  the  meaning  of  the  Proposition  is,  that  the  force 
which  results  out  of  all  tends  to  the  point  S.  But  if  any  force  acts  per 
petually  in  the  direction  of  lines  perpendicular  to  the  described  surface, 
this  force  will  make  the  body  to  deviate  from  the  plane  of  its  motion  :  but 
will  neither  augment  nor  diminish  the  quantity  of  the  described  surface 
and  is  therefore  to  be  neglected  in  the  composition  of  forces. 

PROPOSITION  III.     THEOREM   III. 

Every  body,  that  by  a  radius  drawn  to  the  centre  of  another  body,  how 
soever  moved,  describes  areas  about  that  centre  proportional  to  iJie  times, 
is  urged  by  a  force  compounded  out  of  the  centripetal  force  Bending  fo 
that  other  body,  and  of  all  the  accelerative  force  by  which  that  other 
body  is  impelled. 

Let  L  represent  the  one,  and  T  the  other  body  ;  and  (by  Cor.  0  of  the  Laws) 
if  both  bodies  are  urged  in  the  direction  of  parallel  lines,  by  a  ne T  force 
equal  and  contrary  to  that  by  which  the  second  body  T  is  tinned,  the  first 
body  L  will  go  on  to  describe  about  the  other  body  T  the  same  areas  as 
before  :  but  the  force  by  which  that  other  body  T  was  urged  will  be  now 
destroyed  by  an  equal  and  contrary  force;  and  therefore  (by  Law  I.)  that 
other  body  T,  now  left  to  itself,  will  either  rest,  or  move  uniformly  forward 
in  a  right  line :  and  the  first  body  L  impelled  by  the  difference  of  the 
forces,  that  is,  by  the  force  remaining,  will  go  on  to  describe  about  the  other 
body  T  areas  proportional  to  the  times.  And  therefore  (by  Theor.  II.)  the 
difference  ;f  the  forces  is  directed  to  the  other  body  T  as  its  centre.  Q.E.D 


SEC.  IL]  OF    NATURAL    PHILOSOPHY.  107 

Co.*.  1.  Hence  if  the  one  body  L,  by  a  radius  drawn  to  the  other  body  T, 
describes  areas  proportional  to  the  times  ;  and  from  the  whole  force,  by  which 
the  firr.t  body  L  is  urged  (whether  that  force  is  simple,  or,  according  to 
Cor.  2  of  the  Laws,  compounded  out  of  several  forces),  we  subduct  (by  the 
same  Cor.)  that  whole  accelerative  force  by  which  the  other  body  is  urged  ; 
the  who_e  remaining  force  by  which  the  first  body  is  urged  will  tend  to  the 
( ther  body  T,  as  its  centre. 

COR.  2.  And,  if  these  areas  are  proportional  to  the  times  nearly,  the  re 
maining  force  will  tend  to  the  other  body  T  nearly. 

COR.  3.  And  vice  versa,  if  the  remaining  force  tends  nearly  to  the  other 
body  T,  those  areas  will  be  nearly  proportional  to  the  times. 

COR.  4.  If  the  body  L,  by  a  radius  drawn  to  the  other  body  T,  describes 
areas,  which,  compared  with  the  times,  are  very  unequal ;  and  that  other 
body  T  be  either  at  rest,  or  moves  uniformly  forward  in  a  right  line  :  the 
action  of  the  centripetal  force  tending  to  that  other  body  T  is  either  none 
at  all,  or  it  is  mixed  and  compounded  with  very  powerful  actions  of  other 
forces :  and  the  whole  force  compounded  of  them  all,  if  they  are  many,  is 
directed  to  another  (immovable  or  moveaJble)  centre.  The  same  thing  ob 
tains,  when  the  other  body  is  moved  by  any  motion  whatsoever ;  provided 
that  centripetal  force  is  taken,  wrhich  remains  after  subducting  that  whole 
force  acting  upon  that  other  body  T. 

SCHOLIUM. 

Because  the  equable  description  of  areas  indicates  that  a  centre  is  re 
spected  by  that  force  with  which  the  body  is  most  affected,  and  by  which  it 
is  drawn  back  from  its  rectilinear  motion,  and  retained  in  its  orbit ;  why 
may  we  not  be  allowed,  in  the  following  discourse,  to  use  the  equable  de 
scription  of  areas  as  an  indication  of  a  centre,  about  which  all  circular 
motion  is  performed  in  free  spaces  ? 

PROPOSITION  IV.     THEOREM  IV. 

The  centripetal  forces  of  bodies,  which  by  equable  'motions  describe  differ 
ent  circles,  tend  to  the  centres  of  the  same  circles  ;  and  are  one  to  tJie 
other  as  the  squares  of  t/ie  arcs  described  in  equal  times  applied  to  the 
radii  of  the  circles. 

These  forces  tend  to  the  centres  of  the  circles  (by  Prop.  II.,  and  Cor.  2, 
Prop.  L),  and  are  one  to  another  as  the  versed  sines  of  the  least  arcs  de 
scribed  in  equal  times  (by  Cor.  4,  Prop.  I.) ;  that  is,  as  the  squares  of  the 
same  arcs  applied  to  the  diameters  of  the  circles  (by  Lem.  VII.) ;  and  there 
fore  since  those  arcs  are  as  arcs  described  in  any  equal  times,  and  the  dia- 
me  ers  ace  as  the  radii,  the  forces  will  be  as  the  squares  of  any  arcs  de- 
scr  bed  in  the  same  time  applied  to  the  radii  of  the  circles.  Q.E.D. 
^OR.  1.  Therefore,  since  those  arcs  are  as  the  velocities  of  the  bodies. 


I  OS  THE    MATHEMATICAL    PRINCIPLES  [BOOK    . 

the  centripetal  forces  are  in  a  ratio  compounded  of  the  duplicate  ra'jio  of 
the  velocities  directly,  and  of  the  simple  ratio  of  the  radii  inversely. 

COR.  2.  And  since  the  periodic  times  are  in  a  ratio  compounded  of  the 
ratio  of  the  radii  directly,  and  the  ratio  of  the  velocities  inversely,  the  cen 
tripetal  forces,  are  in  a  ratio  compounded  of  the  ratio  of  the  radii  directly, 
and  the  duplicate  ratio  of  the  periodic  times  inversely. 

COR,  3.  Whence  if  the  periodic  times  are  equal,  and  the  velocities 
therefore  as  the  radii,  the  centripetal  forces  will  be  also  as  the  radii ;  and 
tke  contrary. 

COR.  4.  If  the  periodic  times  and  the  velocities  are  both  in  the  subdu- 
plicate  ratio  of  the  radii,  the  centripetal  forces  will  be  equal  among  them 
selves  ;  and  the  contrary. 

COR.  5.  If  the  periodic  times  are  as  the  radii,  and  therefore  the  veloci 
ties  equal,  the  centripetal  forces  will  be  reciprocally  as  the  radii ;  and  the 
contrary. 

COR.  6.  If  the  periodic  times  are  in  the  sesquiplicate  ratio  of  the  radii, 
and  therefore  the  velocities  reciprocally  in  the  subduplicate  ratio  of  the 
radii,  the  centripetal  forces  will  be  in  the  duplicate  ratio  of  the  radii  in 
versely  :  and  the  contrary. 

COR.  7.  And  universally,  if  the  periodic  time  is  as  any  power  Rn  of  the 
radius  R,  and  therefore  the  velocity  reciprocally  as  the  power  Rn  ]  of 
the  radius,  the  centripetal  force  will  be  reciprocally  as  the  power  R2n  1  of 
the  radius;  and  the  contrary. 

COR.  8.  The  same  things  all  hold  concerning  the  times,  the  velocities, 
and  forces  by  which  bodies  describe  the  similar  parts  of  any  similar  figures 
that  have  their  centres  in  a  similar  position  with  those  figures  ;  as  appears 
by  applying  the  demonstration  of  the  preceding  cases  to  those.  And  the 
application  is  easy,  by  only  substituting  the  equable  description  of  areas  in 
the  place  of  equable  motion,  and  using  the  distances  of  the  bodies  from  the 
centres  instead  of  the  radii. 

COR.  9.  From  the  same  demonstration  it  likewise  follows,  that  the  arc 
which  a  body,  uniformly  revolving  in  a  circle  by  means  of  a  given  centri 
petal  force,  describes  in  any  time,  is  a  mean  proportional  between  the 
diameter  of  the  circle,  and  the  space  which  the  same  body  falling  by  the 
same  given  force  would  descend  through  in  the  same  given  time. 

SCHOLIUM. 

The  case  of  the  6th  Corollary  obtains  in  the  celestial  bodies  (as  Sir 
Christopher  Wren,  Dr.  Hooke,  and  Dr.  Halley  have  severally  observed)  ; 
and  therefore  in  what  follows,  I  intend  to  treat  more  at  large  of  those 
things  which  relate  to  centripetal  force  decreasing  in  a  duplicate  ratio 
of  the  distances  from  the  centres. 

Moreover,  by  means  of  the  preceding  Proposition  and  its  Corollaries,  we 


SEC.  II.]  OF    NATURAL    PHILOSOPHY.  109 

may  discover  the  proportion  of  a  centripetal  force  to  any  other  known 
force,  such  as  that  of  gravity.  For  if  a  body  by  means  of  its  gravity  re 
volves  in  a  circle  concentric  to  the  earth,  this  gravity  is  the  centripetal 
force  of  that  body.  But  from  the  descent  of  heavy  bodies,  the  time  of  one 
entire  revolution,  as  well  as  the  arc  described  in  any  given  time,  is  given 
(by  Cor.  9  of  this  Prop.).  And  by  such  propositions,  Mr.  Huygens,  in  his 
excellent  book  De  Horologio  Oscillatorio,  has  compared  the  force  of 
gravity  with  the  centrifugal  forces  of  revolving  bodies. 

The  preceding  Proposition  may  be  likewise  demonstrated  after  this 
manner.  In  any  circle  suppose  a  polygon  to  be  inscribed  of  any  number 
of  sides.  And  if  a  body,  moved  with  a  given  velocity  along  the  sides  of  the 
polygon,  is  reflected  from  the  circle  at  the  several  angular  points,  the  force, 
with  which  at  every  reflection  it  strikes  the  circle,  will  be  as  its  velocity  : 
and  therefore  the  sum  of  the  forces,  in  a  given  time,  will  be  as  that  ve 
locity  and  the  number  of  reflections  conjunctly ;  that  is  (if  the  species  of 
the  polygon  be  given),  as  the  length  described  in  that  given  time,  and  in 
creased  or  diminished  in  the  ratio  of  the  same  length  to  the  radius  of  the 
circle ;  that  is,  as  the  square  of  that  length  applied  to  the  radius ;  and 
therefore  the  polygon,  by  having  its  sides  diminished  in'inftnitum,  coin 
cides  with  the  circle,  as  the  square  of  the  arc  described  in  a  given  time  ap 
plied  to  the  radius.  This  is  the  centrifugal  force,  with  which  the  body 
impels  the  circle ;  and  to  which  the  contrary  force,  wherewith  the  circle 
continually  repels  the  body  towards  the  centre,  is  equal. 

PROPOSITION  V.     PROBLEM  I. 

There  being  given,  in  any  places,  the  velocity  with  which  a  body  de 
scribes  a  given  figure,  by  means  of  forces  directed  to  some  common 
centre :  to  find  that  centre. 

Let  the  three  right  lines  PT,  TQV,  VR 
touch  the  figure  described  in  as  many  points, 
P,  Q,  R,  and  meet  in  T  and  V.  On  the  tan 
gents  erect  the  perpendiculars  PA,  QB,  RC, 
reciprocally  proportional  to  the  velocities  of  the 
body  in  the  points  P,  Q,  R,  from  which  the 
perpendiculars  were  raised ;  that  is,  so  that  PA 
may  be  to  QB  as  the  velocity  in  Q  to  the  velocity  in  P,  and  QB  to  RC 
as  the  velocity  in  R  to  the  velocity  in  Q.  Through  the  ends  A,  B,  C,  of 
the  perpendiculars  draw  AD,  DBE,  EC,  at  right  angles,  meeting  in  D  and 
E :  and  the  right  lines  TD,  VE  produced,  will  meet  in  S,  the  centre  re 
quired. 

For  the  perpendiculars  let  fall  from  the  centre  S  on  the  tangents  PT. 
QT.  are  reciprocally  as  the  velocities  of  the  bodies  in  the  points  P  and  Q 


110  THE    MATHEMATICAL    PRINCIPLES  [BOOK    1 

(by  Cor.  1,  Prop.  I.),  and  therefore,  by  construction,  as  the  perpendiculars 
AP,  BQ,  directly  ;  that  is,  as  the  perpendiculars  let  fall  from  the  point  D 
on  the  tangents.  Whence  it  is  easy  to  infer  that  the  points  S,  D,  T,  are 
in  one  right  line.  And  by  the  like  argument  the  points  S,  E,  V  are  also 
in  one  right  line  ;  and  therefore  the  centre  S  is  in  the  point  where  the 
right  lines  TD;  YE  meet.  Q.E.D. 

PROPOSITION  VL     THEOREM  V. 

In  a  space  void  of  resistance,  if  a  body  revolves  in  any  orbit  about  an  im 
movable  centre,  and  in  the  least  time  describes  any  arc  just  then,  na 
scent  ;  and  the  versed  sine  of  that  arc  is  supposed  to  be  drawn  bisect 
ing  the  chord,  and  produced  passing  through  the  centre  of  force:  the 
centripetal  force  in  the  middle  of  the  arc  will  be  as  the  versed  sine  di 
rectly  and  the  square  of  the  time  inversely. 
For  the  versed  sine  in  a  given  time  is  as  the  force  (by  Cor.  4,  Prop.  1)  ; 

and  augmenting  the  time  in  any  ratio,  because  the  arc  will  be  augmented 

in  the  same  ratio,  the  versed  sine  will  be  augmented  in  the  duplicate  of 

that  ratio  (by  Cor.  2  and  3,  Lem.  XL),  and  therefore  is  as  the  force  and  the 

square  of  the  time.     Subduct  on  both  sides  the  duplicate  ratio   of  the 

time,  and  the  force  will  be  as  the  versed  sine  directly,  arid  the  square  of 

the  time  inversely.     Q.E.D. 

And  the  same  thing  may  also   be  easily  demonstrated   by  Corol.   4? 

T,em.  X. 

COR.  1.  If  a  body  P  revolving  about  the 

centre  S  describes  a  curve  line  APQ,,  which  a 

right  line  ZPR  touches  in  any  point  P  ;  and 

from  any  other  point  Q,  of  the  curve,  QJl  is 

drawn  parallel  to  the  distance  SP,  meeting 

the  tangent  in  R  ;  and  QT  is  drawn  perpen- 

(licular  to  the  distance  SP  ;  the  centripetal  force  will  be  reciprocally  as  the 

sp2  x  Q/r2 

solid  -  —      :—  ,  if  the  solid  be  taken  of  that  magnitude  which  it  ulti- 


mately  acquires  when  the  points  P  and  Q,  coincide.  For  Q,R  is  equal  to 
the  versed  sine  of  double  the  arc  QP,  whose  middle  is  P  :  and  double  the 
triangle  SQP,  or  SP  X  Q,T  is  proportional  to  the  time  in  which  that 
double  arc  is  described  ;  and  therefore  may  be  used  for  the  exponent  of 
the  time. 

COR.  2.  By  a  like  reasoning,  the  centripetal  force  is  reciprocally  as  the 

SY2  X  QJP2 
solid  -  7^5  -  ;  if  SY  is  a  perpendicular  from  the  centre  of  force  on 


PR  the  tangent  of  the  orbit.     For  the  rectangles  SY  X  QP  and  SP  X  Q,T 
are  equal. 


SEC.  II.]  OF    NATURAL    PHILOSOPHY.  Ill 

COR.  3.  If  the  orbit  is  cither  a  circle,  or  touches  or  cuts  a  circle  c<  ncen- 
trically,  that  is,  contains  with  a  circle  the  least  angle  of  contact  or  sec 
tion,  having  the  same  curvature  rnd  the  same  radius  of  curvature  at  the 
point  P  :  and  if  PV  be  a  chord  of  this  circle,  drawn  from  the  body  through 
the  centre  of  force  ;  the  centripetal  force  will  be  reciprocally  as  the  solid 

QP2 

SY2  X  PV.     For  PV  is       -  . 


COR.  4.  The  same  things  being  supposed,  the  centripetal  force  is  as  the 
square  of  the  velocity  directly,  and  that  chord  inversely.  For  the  velocity 
is  reciprocally  as  the  perpendicular  SY,  by  Cor.  1.  Prop.  I. 

COR.  5.  Hence  if  any  curvilinear  figure  APQ,  is  given,  and  therein  a 
point  S  is  also  given,  to  which  a  centripetal  force  is  perpetually  directed. 
that  law  of  centripetal  force  may  be  found,  by  which  the  body  P  will  bcj 
continually  drawn  back  from  a  rectilinear  course,  and.  being  detained  in 
the  perimeter  of  that  figure,  will  describe  the  same  by  a  perpetual  revolu- 

SP2  x  QT2 

tion.     That  is,  we  are  to  find,  by  computation,  either  the  solid  ----  - 


or  the  solid  SY2  X  PV,  reciprocally  proportional  to  this  force.     Example: 
of  this  we  shall  give  in  the  following  Problems. 

PROPOSITION  VII.     PROBLEM  II. 

Tf  a  body  revolves  in  the  circumference  of  a  circle;  it  is  proposed  to  finii 
the  law  of  centripetal  force  directed  to  any  given,  point. 

Let  VQPA  be  the  circumference  of  the 
circle  ;  S  the  given  point  to  which  as  to 
a  centre  the  force  tends  :  P  the  body  mov 
ing  in  the  circumference  ;  Q  the  next 
place  into  which  it  is  to  move;  and  PRZ 
the  tangent  of  the  circle  at  the  preceding 
place.  Through  the  point  S  draw  the  v 
chord  PV,  and  the  diameter  VA  of  the 
circle  :  join  AP,  and  draw  Q,T  perpen 
dicular  to  SP,  which  produced,  may  meet 
the  tangent  PR  in  Z  ;  and  lastly,  through 
the  point  Q,  draw  LR  parallel  to  SP,  meeting  the  circle"  in  L,  and  the 
tangent  PZ  in  R.  And,  because  of  the  similar  triangles  ZQR,  ZTP. 
VPA,  we  shall  have  RP2,  that  is.  QRL  to  QT2  as  AV2  to  PV2.  And 

QRlj  x  PV2  SI3-' 

therefore  '-    —TS  --  is  equal  to  QT2.     Multiply  those  equals  by     -'. 


and  the  points  P  and  Q,  coinciding,  for  RL  write  PV  ;  then  we  shall  have 

SP-'  X  PV5       SP2  x  QT2 

—  •     And  therefore  fl»r  Cor  1  and  5.  Prop.  VI.) 


112  THE   MATHEMATICAL    PRINCIPLES  [BOOK    I, 

SP2  X  PV3 

the  centripetal  force  is  reciprocally  as  -  —  ry^~  —  J  that  is  (because  AV2 


ia  given),  reciprocally  as  the  square  of  the  distance  or  altitude  SP,  and  the 
3ube  of  the  chord  PV  conjunctly.     Q.E.L 

The  same  otherwise. 

On  the  tangent  PR  produced  let  fall  the  perpendicular  SY  ;  and  (be 
cause  of  the  similar  triangles  SYP,  VPA),  we  shall  have  AV  to  PV  as  SP 

SP  X  PV  SP2  ><  PV3 

to  SY,  and  therefore  --  ^~—  -  =  SY,  and  -    —  ^-  =  SY2  X  PV. 
A  V  A  V 

And  therefore  (by  Corol.  3  and  5,  Prop.  VI),  the  centripetal  force  is  recip- 

SP2  X  PV3 

rocally  as  -  ~~ry¥~~~  I  *na*  *s  (because  AV  is  given),  reciprocally  as  SP" 

X  PV3.     Q.E.I. 

Con.  1.  Hence  if  the  given  point  S,  to  which  the  centripetal  force  al 
ways  tends,  is  placed  in  the  circumference  of  the  circle,  as  at  V,  the  cen 
tripetal  force  will  be  reciprocally  as  the  quadrato-cube  (or  fifth  power)  of 
the  altitude  SP. 

COR.  2.  The  force  by  which  the  body  P  in  the 
circle  APTV  revolves  about  the  centre  of  force  S 
is  to  the  force  by  which  the  same  body  P  may  re 
volve  in  the  same  circle,  and  in  the  same  periodic 
time,  about  any  other  centre  of  force  R,  as  RP2  X 
SP  to  the  cube  of  the  right  line  SG,  which,  from 
the  first  centre  of  force  S  is  drawn  parallel  to  the 
distance  PR  of  the  body  from  the  second  centre  of  force  R,  meeting  the 
tangent  PG  of  the  orbit  in  G.  For  by  the  construction  of  this  Proposition, 
the  former  force  is  to  the  latter  as  RP2  X  PT3  to  SP2  X  PV3;  that  is,  as 

SP3  X  PV3 

SP  X  RP2  to  --  p™  —  ;  or  (because  of  the  similar  triangles  PSG,  TPV) 

to  SGS. 

COR.  3.  The  force  by  which  the  body  P  in  any  orbit  revolves  about  the 
centre  of  force  S,  is  to  the  force  by  which  the  same  body  may  revolve  in 
the  same  orbit,  and  the  same  periodic  time,  about  any  other  centre  of  force 
R.  as  the  solid  SP  X  RP2,  contained  under  the  distance  of  the  body  from 
the  first  centre  of  force  S,  and  the  square  of  its  distance  from  the  sec 
ond  centre  of  force  R,  to  the  cube  of  the  right  line  SG,  drawn  from  the 
first  centre  of  the  force  S,  parallel  to  the  distance  RP  of  the  body  from 
fch*3  second  centre  of  force  R,  meeting  the  tangent  PG  of  the  orbit  in  G. 
For  the  force  in  this  orbit  at  any  point  P  is  the  same  as  in  a  circle  of  the 
same  curvature. 


SJSG.  IL] 


OF    NATURAL    PHILOSOPHY. 


113 


PROPOSITION  VIII.     PROBLEM  III. 

If  a  body  mi  ues  in  the  semi-circuwferencePQA:  it  is  proposed  to  find 
the  law  of  the  centripetal  force  tending  to  a  point  S,  so  remote,  that  all 
the  lines  PS.  RS  drawn  thereto,  may  be  taken  for  parallels. 
From  C,  the  centre  of  the  semi-circle,  let 

the  semi-diameter  CA  he  drawn,  cutting  the 

parallels  at  right  angles  in  M  and  N,  and 

join  CP.     Because  of  the  similar  triangles 

CPM,  PZT,  and  RZQ,  we  shall  have  CP2 

to  PM2  as  PR2  to  QT2;  and,  from  the  na 

ture  of  the  circle,  PR2  is  equal  to  the  rect 

angle  QR  X  RN  +  QN,  or,  the  points  P,  Q  coinciding,  to  the  rectangle 

QR  x  2PM.     Therefore  CP2  is  to  PM2  as  QR  X  2PM  to  QT2;  and 

QT2        2PM3  QT2  X  SP2        2PM3  X  SP2 

therefore  (by 


QR 

Corol. 
8PM3  X  SP2 


, 
and 


QR 


And 


1    and   5,   Prop.   VI.),    the   centripetal   force   is   reciprocally   as 

2SP2 


. 
that  is  (neglecting  the  given  ratio  -ppr)>  reciprocally  as 

PM3.     Q.E.L 

And  the  same  thing  is  likewise  easily  inferred  from  the  preceding  Pro 
position. 

SCHOLIUM. 

And  by  a  like  reasoning,  a  body  will  be  moved  in  an  ellipsis,  or  even  ia 
an  hyperbola,  or  parabola,  by  a  centripetal  force  which  is  reciprocally  ae 
the  cube  of  the  ordinate  directed  to  an  infinitely  remote  centre  of  force. 

PROPOSITION  IX.     PROBLEM   IV. 

If  a  body  revolves  in  a  spiral  PQS,  cutting  all  the  radii  SP,  SQ,  fyc., 
in  a  given  angle;  it  is  proposed  to  find  thelaio  of  the  centripetal  force 
tending  to  tJie  centre  of  that  spiral. 
Suppose  the  inde 

finitely   small    angle  AY 

PSQ  to  be  given  ;  be 

cause,    then,    all    the 

angles  are  given,  the 

figure  SPRQT   will    ,  _ 

be    given    in   specie.  v 

QT  Q,T2 

Therefore  the  ratio  -7^-  is  also  given,  and    „„    is  as  QT,  that  is  (be 
lot  IX  QK 

cause  the  figure  is  given  in  specie),  as  SP.     But  if  the  angle  PSQ  is  any 
way  changed,  the  right  line  QR,  subtending  the  angle  of  contact  QPU 


tU 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK    J 


(by  Lemma  XI)  will  be  changed  in  the  duplicate  ratio  of  PR  or  QT 

QT2 

Therefore  the  ratio  ~TVD~ remains  the  same  as  before,  that  is,  as  SP.   And 

QT2  x  SP2 

— -^ is  as  SP3,  and  therefore  (by  Corol.  1  and  5,  Prop.  YI)  the 

centripetal  force  is  reciprocally  as  the  cube  of  the  distance  SP.     Q.E.I. 

The  same  otherwise. 

The  perpendicular  SY  let  fall  upon  the  tangent,  and  the  chord  PY  of 
the  circle  concentrically  cutting  the  spiral,  are  in  given  ratios  to  the  height 
SP ;  and  therefore  SP3  is  as  SY2  X  PY,  that  is  (by  Corol.  3  and  5,  Prop. 
YI)  reciprocally  as  the  centripetal  force. 

LEMMA  XII. 

All  parallelograms  circumscribed  about  any  conjugate  diameters  of  a 
given  ellipsis  or  hyperbola  are  equal  among  themselves. 
This  is  demonstrated  by  the  writers  on  the  conic  sections. 

PROPOSITION  X.     PROBLEM  Y. 

If  a  body  revolves  in  an  ellipsis  ;  it  is  proposed  to  find  the  law  of  thi 
centripetal  force  tending  to  the  centre  of  the  ellipsis. 

Suppose  CA,  CB  to 
be  semi-axes  of  the 
ellipsis;  GP, DK, con 
jugate  diameters ;  PF, 
Q,T  perpendiculars  to 
those  diameters;  Qvan 
^rdinate  to  the  diame 
ter  GP ;  and  if  the 
parallelogram  QvPR 
be  completed,  then  (by 
the  properties  of  the 
jonic  sections)  the  rec- 
langle  PvG  will  be  to 
Qv2  as  PC2  to  CD2; 
and  (because  of  the 
similar  triangles  Q^T,  PCF),  Qi>2  to  QT2  as  PC2  to  PF2 ;  and,  by  com 
position,  the  ratio  of  PvG  to  QT2  is  compounded  of  the  ratio  of  PC2 1< 

QT2 
CD2,    and   of    the    ratio  of    PC2   to  PF2,  that   is,   vG   to  -p—  as  PC; 

to_92L^_P_]^_.     Put  QR  for  Pr,  and  (by  Lem.  XII)  BC  X  CA  for  CD 

K  PF ;  also  (the  points  P  and  Q  coinciding)  2PC  for  rG;  and  multiply- 


SEC.  II.]  OF    NATURAL    PHILOSOPHY.  115 

QT2  x  PC2 

ing  the  extremes  and  means  together,  we  shall  have rfo~ equal  to 

2BC2  X  CA2 

— pp —  — .     Therefore  (by  Cor.  5,  Prop.  VI),  the  centripetal  force  is 

2BC2  X  CA2 

reciprocally  as  —    — ry~ ;    that  is  (because  2I3C2  X  CA2  is  given),  re 
ciprocally  as-r^v;  that  is,  directly  as  the  distance  PC.     QEI. 
I  O 

TJie  same  otherwise. 

[n  the  right  line  PG  on  the  other  side  of  the  point  T,  take  the  point  u 
so  that  Tu  may  be  equal  to  TV  ;  then  take  uV,  such  as  shall  be  to  v G  as 
DC2  to  PC2.  And  because  Qr9  is  to  PvG  as  DC'2  to  PC2  (by  the  conic 
sections),  we  shall  have  Qv2  •-=  Pi'  X  «V.  Add  the  rectangle  n.Pv  to  both 
sides,  and  the  square  of  the  chord  of  the  arc  PQ,  will  be  equal  to  the  rect 
angle  VPv ;  and  therefore  a  circle  which  touches  the  conic  section  in  P, 
and  passes  through  the  point  Q,,  will  pass  also  through  the  point  V.  Now 
let  the  points  P  and  Q,  meet,  and  the  ratio  of  nV  to  rG,  which  is  the  same 
with  the  ratio  of  DC2  to  PC2,  will  become  the  ratio  of  PV  to  PG,  or  PV 

2DC2 

to  2PC  :  and  therefore  PY  will  be  equal  to    „„ — .     And    therefore    the 

force  by  which  the  body  P  revolves  in  the  ellipsis  will  be  reciprocally  as 

2  DC2 

— ry—  X  PF2  (by  Cor.  3,  Prop.  VI) ;    that  is  (because  2DC2  X  PF2  is 
I  O 

given)  directly  as  PC.     Q.E.I. 

COR.  1.  And  therefore  the  force  is  as  the  distance  of  the  body  from  the 
centre  of  the  ellipsis ;  and,  vice  versa,  if  the  force  is  as  the  distance,  the 
body  will  move  in  an  ellipsis  whose  centre  coincides  with  the  centre  of  force, 
or  perhaps  in  a  circle  into  which  the  ellipsis  may  degenerate. 

COR.  2.  And  the  periodic  times  of  the  revolutions  made  in  all  ellipses 
whatsoever  about  the  same  centre  will  be  equal.  For  those  times  in  sim 
ilar  ellipses  will  be  equal  (by  Corol.  3  and  S,  Prop.  IV) ;  but  in  ellipses 
that  have  their  greater  axis  common,  they  are  one  to  another  as  the  whole 
areas  of  the  ellipses  directly,  and  the  parts  of  the  areas  described  in  the 
same  time  inversely:  that  is,  as  the  lesser  axes  directly,  and  the  velocities 
of  the  bodies  in  their  principal  vertices  inversely ;  :hat  is,  as  those  lesser 
axes  dirtily,  and  the  ordinates  to  the  same  point  %f  the  common  axes  in 
versely  ;  and  therefore  (because  of  the  equality  of  the  direct  and  inverse 
ratios)  in  the  ratio  of  equality. 

SCHOLIUM. 

If  the  ellipsis,  by  having  its  centre  removed  to  an  infinite  distance,  de 
generates  into  a  parabola,  the  body  will  move  in  tin's  parabola ;  and  the 


116 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK  I 


force,  now  tending  to  a  centre  infinitely  remote,  will  become  equable. 
Which  is  Galileo  s  theorem.  And  if  the  parabolic  section  of  the  cone  (by 
changing  the  inclination  of  the  cutting  plane  to  the  cone)  degenerates  into 
an  hyperbola,  the  body  will  move  in  the  perimeter  of  this  hyperbola,  hav 
ing  its  centripetal  force  changed  into  a  centrifugal  force.  And  in  like 
manner  as  in  the  circle,  or  in  the  ellipsis,  if  the  forces  are  directed  to  the 
centre  of  the  figure  placed  in  the  abscissa,  those  forces  by  increasing  or  di 
minishing  the  ordinates  in  any  given  ratio,  or  even  by  changing  the  angle 
of  the  inclination  of  the  ordinates  to  the  abscissa,  are  always  augmented 
or  diminished  in  the  ratio  of  the  distances  from  the  centre ;  provided  the 
periodic  times  remain  equal ;  so  also  in  all  figures  whatsoever,  if  the  ordi- 
nates  are  augmented  or  diminished  in  any  given  ratio,  or  their  inclination 
is  any  way  changed,  the  periodic  time  remaining  the  same,  the  forces  di 
rected  to  any  centre  placed  in  the  abscissa  are  in  the  several  ordinatee 
augmented  or  diminished  in  the  ratio  of  the  distances  from  the  centre 


SECTION  III. 

Of  the  motion  of  bodies  in  eccentric  conic  sections. 

PROPOSITION  XL     PROBLEM  VI. 

If  a  body  revolves  in  an  ellipsis  ;  it  is  required  to  find  the  law  of  the 

centripetal  force  tending  to  the  focus  of  the  ellipsis. 

Let  S  be  the  focus 
of  the  ellipsis.  Draw 
SP  cutting  the  diame 
ter  DK  of  the  ellipsis 
in  E,  and  the  ordinate 
Qv  in  x ;  and  com 
plete  the  parallelogram 
d.rPR,  It  is  evident 
that  EP  is  equal  to  the 
greater  semi-axis  AC : 
for  drawing  HI  froln 
the  other  focus  H  of 
the  ellipsis  parallel  to 
EC,  because  CS,  CH 
are  equal,  ES,  El  will 
be  also  equal ;  so  that  EP  is  the  half  sum  of  PS,  PI,  that  is  (because  of 
the  parallels  HI,  PR,  and  the  equal  angles  IPR,  HPZ),  of  PS,  PH,  which 
taken  together  are  equal  to  the  whole  axis  2AC.  Draw  QT  perpendicu 
lar  to  SP,  and  putting  L  for  the  princi  al  latus  rectum  of  the  ellipsis  (or  for 


III.  OF    NATURAL    PHILOSOPHY.  117 


L    X    ^R  t0  L    X    Py  aS  ^R  t0  PV   that  1S>  US  PE 


or  AC  to  PC  ;  and  L  X  Pv  to  GvP  as  L  to  Gy  ;  and  GvP  to  Qi>2  as 
to  CD-  ;  and  by  (Corol.  2,  Lem.  VII)  the  points  Q,  and  P  coinciding,  Qv* 
is  to  Q,r-  in  the  ratio  of  equality  ;  and  Q,.r2  or  Qv2  is  to  Q,T2  as  EP2  to 
PF2,  that  is,  as  CA2  to  PF2,  or  (by  Lem.  XII)  as  CD'2  to  CB2.  And  com 
pounding  all  those  ratios  together,  we  shall  have  L  X  QR  to  Q,T2  as  AC 
X  L  X  PC2  X  CD2,  or  2CB2  X  PC2  X  CD2  to  PC  X  Gv  X  CD2  X 
CB2,  or  as  2PC  to  Gv.  But  the  points  Q  and  P  coinciding,  2PC  and  Gr 
are  equal.  And  therefore  the  quantities  L  X  QR  and  Q,T2,  proportional 

SP2 

to  these,  will  be  also  equal.     Let  those  equals  be  drawn  into-p^B"?  and  L 


SP2  X  QT2 

X  SP2  will  become  equal  to  --  ^p  —  —  .    And  therefore  (by  Corol.  1  and 


5,  Prop.  VI)  the  centripetal  force  is  reciprocally  as  L  X  SP2,  that  is,  re 
ciprocally  in  the  duplicate  ratio  of  the  distance  SP.     Q.E.I. 

The  same  otherwise. 

Since  the  force  tending  to  the  centre  of  the  ellipsis,  by  which  the  body 
P  may  revolve  in  that  ellipsis,  is  (by  Corol.  1,  Prop.  X.)  as  the  distance 
CP  of  the  body  from  the  centre  C  of  the  ellipsis  ;  let  CE  be  drawn  paral 
lel  to  the  tangent  PR  of  the  ellipsis  :  and  the  force  by  which  the  same  body 
P  may  revolve  about  any  other  point  S  of  the  ellipsis,  if  CE  and  PS  in- 

PE3 

tersect  in  E,  will  be  as  ^T3,  (by  Cor.  3,  Prop.  VII.)  ;  that  is,  if  the  point 

S  is  the  focus  of  the  ellipsis,  and  therefore  PE  be  given  as  SP2  recipro 
cally.     Q.E.I. 

With  the  same  brevity  with  which  we  reduced  the  fifth  Problem  to  the 
parabola,  and  hyperbola,  we  might  do  the  like  here  :  but  because  of  the 
dignity  of  the  Problem  and  its  use  in  what  follows,  I  shall  confirm  the  other 
cases  by  particular  demonstrations. 

PROPOSITION  XII.     PROBLEM  VII. 

Suppose  a  body  to  move  in  an  hyperbola  ;  it  is  required  to  find  lite  law  of 

the  centripetal  force  tending  to  the  focus  of  that  figure. 

Let  CA,  CB  be  the  semi-axes  of  the   hyperbola  ;  PG,  KD  other  con 

jugate  diameters  ;   PF  a  perpendicular  to  the  diameter  KD  ;  and  Qv  an 

ordinate  to  the  diameter  GP.     Draw  SP  cutting  the  diameter  DK  in  E, 

and  the  ordinate  Qv  in  x,  and  complete  the  parallelogram  QRP.r.     It  is 

evident  that  EP  is  equal   to   the  semi-transverse  axis  AC  ;  for  drawing 

HE,  from  the  other  focus  H  of  the  hyperbola,  parallel  to  EC,  because  CS, 

TH  are  equal,  ES  El  will  be  also  equal  ;  so  that  EP  is  the  half  difference 


J1S 


THE    MATHEMATICAL    PRINCIPLES 


[Book  I 


.of  PS,  PI;  that  is  (be 
cause  of  the  parallels  IH, 
PR,  and  the  equal  angles 
IPR,  HPZ),  of  PS,  PH, 
the  difference  of  which  is 
equal  to  the  whole  axis 
2AC.  Draw  Q,T  perpen 
dicular  to  SP;  and  put 
ting  L  for  the  principal 
latus  rectum  of  the  hy 
perbola  (that  is,  for 

2BC2\  .... 

-Tp-  )  7  we  shall  have  L 

X  QR  to  L  X  Pv  as  QR 
to  Pv,  or  Px  to  Pv,  that  is 
(because  of  the  similar  tri 
angles  Pxv,  PEC),  as  PE 
to  PC,  or  AC  to  PC. 
And  L  X  Pv  will  be  to 
Gv  X  Pv  as  L  to  Gv; 
and  (by  the  properties  of 
the  conic  sections)  the  rec 
tangle  G?'P  is  to  Q,v2  as 
PC2  to  CD2 ;  and  by  (Cor.  2,  Lem.  VII.),  Qv2  to  Qa*  the  points  Q  and  P 
coinciding,  becomes  a  ratio  of  equality  ;  and  Q,.r2  or  Qv2  is  to  Q,T2  as  EP2 
to  PF2,  that  is,  as  CA2  to  PF2,  or  (by  Lem.  XII.)  as  CD2  to  CB2 :  and, 
compounding  all  those  ratios  together,  we  shall  have  L  X  Q,R  to  Q,T2  as 
AC  X  L  X  PC2  X  CD2,  or  2CB2  X  PC2  X  CD2  to  PC  X  Gv  X  CD2 
X  CB2,  or  as  2PC  to  Gv.  But  the  points  P  and  Q,  coinciding.  2PC  and 
Gv  are  equal.  And  therefore  the  quantities  L  X  Q,R  arid  Q.T2,  propor 
tional  to  them,  will  be  also  equal.  Let  those  equals  be  drawn  into 

SP2  sp2  x  o/r2 

^,  and  we  shall  have  L  X  SP2  equal  to ^^ .    And  therefore  (by 


Cor.  1.  and  5,  Prop.  VI.)  the  centripetal  force  is  reciprocally  as  L  X  SP'2. 
'hat  is,  reciprocally  in  the  duplicate  ratio  of  the  distance  SP.     Q,.E.I. 

TJie  same  otherwise. 

Find  out  the  force  tending  from  the  centre  C  of  the  hype rbola.     This  will 
be  proportional  to  the  distance  CP.     But  from  thence  (by  Cor.  3,  Prop. 

PE3 
VII.)  the  force  tending  to  the  focus  S  will  be  as  -^-^  th;  (t  is,  because  PE 

is  given  reciprocally  as  SP-.     Q,.E.I. 


SEC.  III.] 


OF    NATURAL    PHILOSOPHY. 


119 


And  the  same  way  may  it  be  demonstrated,  that  the  body  having  its  cen 
tripetal  changed  into  a  centrifugal  force,  will  move  in  the  conjugate  hy 
perbola. 

LEMMA  XIII. 

The  latus  rectum  of  a  parabola  belonging  to  any  vertex  is  quadruple 
the  distance  of  that  vertex  from  the  focus  of  thejigurc. 

This  is  demonstrated  by  the  writers  on  the  conic  sections. 

LEMMA  XIV. 

Tlie  perpendicular,  let  fall  from  the  focus  of  a  parabola  on  its  tangent,  is 
a  mean  proportional  between  the.  distances  of  the  focus  from  the  poini 
of  contact,  and  from  the  principal  vertex  of  the  figure. 

For,  let  AP  be  the  parabola,  S  its 
focus,  A  its  principal  vertex,  P  the 
point  of  contact,  PO  an  ordinate  to  the 
principal  diameter.  PM  the  tangent 
meeting  the  principal  diameter  in  M. 

and  SN  the  perpendicular  from  the  fo-~       M  A       s        o 

cus  on  the  tangent :  join  AN,  and  because  of  the  equal  lines  MS  and  SP, 
MN  and  NP,  MA  and  AC,  the  right  lines  AN,  OP,  will  be  parallel ;  and 
thence  the  triangle  SAN  will  be  right-angled  at  A,  and  similar  to  the 
equal  triangles  SNM,  SNP  j  therefore  PS  is  to  SN  as  SN  to  SA.  Q.E.D. 

COR.  1.  PS2  is  to  SN2  as  PS  to  SA. 

COR.  2.  And  because  SA  is  given,  SN-  will  be  as  PS. 

COR.  3.  And  the  concourse  of  any  tangent  PM,  with  the  right  line  SN. 
drawn  from  the  focus  per]  endicular  on  the  tangent,  falls  in  the  right  line 
AN  that  touches  the  parabola  in  the  principal  vertex. 

PROPOSITION  XIII.     PROBLEM  VIII. 

If  a  body  moves  in  the  perimeter  of  a  parabola  ;  it  is  required  to  find  the. 

law  of  the  centripetal  force  tending  to  the  focus  of  that  figure. 

Retaining  the  construction 
of  the  preceding  Lemma,  let  P 
be  the  body  in  the  perimeter 
of  the  parabola  ;  and  from  the 
place  Q,,  into  which  it  is  next 
to  succeed,  draw  QH  parallel  IS!. 

and  Q,T  perpendicular  to  SP, 
as  also  Qv  parallel  to  the  tan 
gent,  and  mating  the  diame 
ter  PG  in  v,  and  the  distance  — 


120  THE    MATHEMATICAL    PRINCIPLES  [BOOK  I. 

SP  in  x.  Now.  because  of  the  similar  triangles  Pxv,  SPM,  and  of  the 
equal  sides  SP,  SM  of  the  one,  the  sides  Px  or  Q,R  and  Pv  of  the  other 
will  be  also  equal.  But  (by  the  conic  sections)  the  square  of  the  ordinate 
Q,y  is  equal  to  the  rectangle  under  the  latus  rectum  and  the  segment  Pv 
of  the  diameter ;  that  is  (by  Lem.  XIII.),  to  the  rectangle  4PS  X  Pv,  or 
4PS  X  Q,R ;  and  the  points  P  and  Q,  coinciding,  the  ratio  of  Qv  to  Q,.r 
(by  Cor.  2,  Lem.  VII.,)  becomes  a  ratio  of  equality.  And  therefore  Q,#2,  in 
this  case,  becomes  equal  to  the  rectangle  4PS  X  Q,R.  But  (because  of  the 
similar  triangles  Q#T,  SPN),  Q^'2  is  to  QT2  as  PS2  to  SN2,  that  is  (by 
Cor.  1,  Lem.  XIV.),  as  PS  to  SA  ;  that  is,  as  4PS  X  QR  to  4SA  x  QR, 
and  therefore  (by  Prop.  IX.  Lib.  V.,  Elem.)  QT*  and  4SA  X  QR  are 

SP2          SP2  X  QT2 

equal.     Multiply  these  equals  by  ^-^-,  and ^5 — -will  become    equal 

to  SP2  X  4SA :  and  therefore  (by  Cor.  1  and  5,  Prop.  VL),  the  centripetal 
force  is  reciprocally  as  SP2  X  4S A ;  that  is,  because  4SA  is  given,  recipro 
cally  in  the  duplicate  ratio  of  the  distance  SP.  Q.E.I. 

COR.  1.  From  the  three  last  Propositions  it  follows,  that  if  any  body  P 
goes  from  the  place  P  with  any  velocity  in  the  direction  of  any  right  line 
PR,  and  at  the  same  time  is  urged  by  the  action  of  a  centripetal  force  that 
is  reciprocally  proportional  to  the  square  of  the  distance  of  the  places  from 
the  centre,  the  body  will  move  in  one  of  the  conic  sections,  having  its  fo 
cus  in  the  centre  of  force ;  and  the  contrary.  For  the  focus,  the  point  of 
contact,  and  the  position  of  the  tangent,  being  given,  a  conic  section  may 
be  described,  which  at  that  point  shall  have  a  given  curvature.  But  the 
curvature  is  given  from  the  centripetal  force  and  velocity  of  the  body  be 
ing  given ;  and  two  orbits,  mutually  touching  one  the  other,  cannot  be  de 
scribed  by  the  same  centripetal  force  and  the  same  velocity. 

COR.  2.  If  the  velocity  with  which  the  body  goes  from  its  place  P  is 
such,  that  in  any  infinitely  small  moment  of  time  the  lineola  PR  may  be 
thereby  describe  I:  and  the  centripetal  force  such  as  in  the  same  time  to 
move  the  same  body  through  the  space  QR  ;  the  body  will  move  in  one  of 

QT2. 

the  conic  sections,  whose  principal  latus  rectum  is  the  quantity  Tjfr  in  its 

ultimate  state,  when  thelineoke  PR,  QR  are  diminished  in  infinitum.  In 
these  Corollaries  I  consider  the  circle  as  an  ellipsis ;  and  I  except  the  case 
where  the  body  descends  to  the  centre  in  a  right  line. 

PROPOSITION  XIV.     THEOREM  VI. 

Tf  several  bodies  revolve  about  one  common  centre,  and  the  centripetal 
force  is  reciprocally  in  tlie  duplicate  ratio  of  the  distance  of  places 
from  the  centre  ;  I  say,  that  the  principal  latera  recta  of  tfieir  orbits 
are  in  the  duplicate  ratio  of  the  areas,  which  the  bodies  by  radii  drawn 
to  the  centre  describe  it\  the  same  time. 


SEC.  HI.  OF    NATURAL    PHILOSO1  HY. 


For  (by  Cor  2,  Prop.  XIII)  the  latus  rectum 

QT*. 

L  is  equal  to  the  quantity-^-in  its  ultimate 


state  when  the  points  P  and  Q,  coincide.  But 
the  lineola  QR  in  a  given  time  is  as  the  gen 
erating  centripetal  force  ;  that  is  (by  supposi- 

QT2 

tion),  reciprocally  as  SP2.     And  therefore-^-^ 

is  as  Q.T2  X  SP2  ;  that  is,  the  latus  rectum  L  is  in  the  duplicate  ratio  of 
the  area  QT  X  SP.     Q.E.D. 

C?OR.  Hence  the  whole  area  of  the  ellipsis,  and  the  rectangle  under  the 
axes,  which  is  proportional  to  it,  is  in  the  ratio  compounded  of  the  subdu- 
plicate  ratio  of  the  latus  rectum,  and  the  ratio  of  the  periodic  time.  For 
the  whole  area  is  as  the  area  QT  X  SP,  described  in  a  given  time,  mul 
tiplied  by  the  periodic  time. 

PROPOSITION  XV.    THEOREM  VII. 

The  same  things  being  supposed,  J  say,  that  the  periodic  times  in  ellip 
ses  are  in  the  sesquiplicate  ratio  of  their  greater  axes. 
For  the  lesser  axis  is  a  mean  proportional  between  the  greater  axis  and 
the  latus  rectum  ;  and,  therefore,  the  rectangle  under  the  axes  is  in  the 
ratio  compounded  of  the  subduplicate  ratio  of  the  latus  rectum  and  the 
sesquiplicate  ratio  of  the  greater  axis.  But  this  rectangle  (by  Cor.  o. 
Prop.  XIV)  is  in  a  ratio  compounded  of  the  subduplicate  ratio  of  the 
latus  rectum,  and  the  ratio  of  the  periodic  time.  Subduct  from  both  sides 
the  subduplicate  ratio  of  the  latus  rectum,  and  there  will  remain  the  ses 
quiplicate  ratio  of  the  greater  axis,  equal  to  the  ratio  of  the  periodic  time. 
Q.E.D. 

COR.  Therefore  the  periodic  times  in  ellipses  are  the  same  as  in  circles 
whose  diameters  are  equal  to  the  greater  axes  of  the  ellipses. 

PROPOSITION  XVI.     THEOREM  VIII. 

The  same  things  being  supposed,  and  right  lines  being  drawn  to  the 
bodies  that  shall  touch  the  orbits,  and  perpendiculars  being  let  fall  on 
those  tangents  from  the  common  focus  ;  I  say,  that  the  velocities  oj 
the  bodies  are  in  a  ratio  compounded  of  the  ratio  of  the  perpendiculars 
inversely,  and  the,  subduplicate  ratio  of  the  principal  later  a  recta 
direct!]). 
From  the  focus  S  draw  SY  perpendicular  to  the  tangent  PR,  and  the 

velocity  of  the  body  P  will  be  reciprocally  in  the  subduplicate  ratio  of  the 

SY2 
quantity  -y—  .     For   that  velocity  is  as  the  infinitely  small  arc  PQ   de- 


122  THE  MATHEMATICAL    PRINCIPLES  [BOOK    I. 

scribed  in  a  given  moment  of  time,  that  is  (by 
Lem.  "VII),  as  the  tangent  PR ;  that  is  (because 
of  the  proportionals  PR  to  Q,T,  and  SP  to 

SP  X   Q,T 

SY),  as ~y —  — ;   or  as  SY  reciprocally, 

and  SP  X  Q,T  directly  ;  but  SP  X  QT  is  as 
the  area  described  in  the  given  time,  that  is  (by 
Prop.  XIV),  in  the  subduplicate  ratio  of  the 
latus  rectum.  Q.E.D. 

COR.  1.  The  principal  latera  recta  are  in  a  ratio  compounded  of  the 
duplicate  ratio  of  the  perpendiculars  and  the  duplicate  ratio  of  the  ve 
locities. 

COR.  2.  The  velocities  of  bodies,  in  their  greatest  and  least  distances  from 
the  common  focus,  are  in  the  ratio  compounded  of  the  ratio  of  the  distan 
ces  inversely,  and  the  subduplicate  ratio  of  the  principal  latera  recta  di 
rectly.  For  those  perpendiculars  are  now  the  distances. 

COR.  3.  Arid  therefore  the  velocity  in  a  conic  section,  at  its  greatest  or 
least  distance  from  the  focus,  is  to  the  velocity  in  a  circle,  at  the  same  dis 
tance  from  the  centre,  in  the  subduplicate  ratio  of  the  principal  latus  rec 
tum  to  the  double  of  that  distance. 

COR.  4.  The  velocities  of  the  bodies  revolving  in  ellipses,  at  their  mean 
distances  from  the  common  focus,  are  the  same  as  those  of  bodies  revolving 
in  circles,  at  the  same  distances  ;  that  is  (by  Cor.  6.  Prop.  IV),  recipro 
cally  in  the  subduplicate  ratio  of  the  distances.  For  the  perpendiculars 
are  now  the  lesser  semi-axes,  and  these  are  as  mean  proportionals  between 
the  distances  and  the  latera  recta.  Let  this  ratio  inversely  be  compounded 
with  the  subduplicate  ratio  of  the  latera  recta  directly,  and  we  shall  have 
the  subduplicate  ratio  of  the  distance  inversely. 

COR.  5.  In  the  same  figure,  or  even  in  different  figures,  whose  principal 
latera  recta  are  equal,  the  velocity  of  a  body  is  reciprocally  as  the  perpen 
dicular  let  fall  from  the  focus  on  the  tangent. 

COR.  6.  In  a  parabola,  the  velocity  is  reciprocally  in  the  subduplicate 
ratio.of  the  distance  of  the  body  from  the  focus  of  the  figure;  it  is  more 
variable  in  the  ellipsis,  and  less  in  the  hyperbola,  than  according  to  this 
ratio.  For  (by  Cor.  2,  Lem.  XIV)  the  perpendicular  let  fall  from  the 
focus  on  the  tangent  of  a  parabola  is  in  the  subduplicate  ratio  of  the  dis 
tance.  In  the  hyperbola  the  perpendicular  is  less  variable  ;  in  the  ellipsis 
more. 

COR.  7.  In  a  parabola,  the  velocity  of  a  body  at  any  distance  from  the 
focus  is  to  the  velocity  of  a  body  revolving  in  a  circle,  at  the  same  distance 
from  the  centre,  in  the  subduplicate  ratio  of  the  number  2  to  1 ;  in  the 
ellipsis  it  is  less,  and  in  the  hyperbola  greater,  than  according  to  this  ratio, 
For  (by  Cor.  2  of  this  Prop.)  the  velocitv  at  the  vertex  of  a  parabola  is  ir 


SEC.  III.] 


OF    NATURAL    PHILOSOPHY. 


123 


this  ratio,  and  (by  Cor.  6  of  this  Prop,  and  Prop.  IV)  the  same  proportion 
holds  in  all  distances.  And  hence,  also,  in  a  parabola,  the  velocity  is 
everywhere  equal  to  the  velocity  of  a  body  revolving  in  a  circle  at  half  the 
distance ;  in  the  ellipsis  it  is  less,  and  in  the  hyperbola  greater. 

COR.  S.  The  velocity  of  a  body  revolving  in  any  conic  section  is  to  the 
velocity  of  a  body  revolving  in  a  circle,  at  the  distance  of  half  the  princi 
pal  latus  rectum  of  the  section,  as  that  distance  to  the  perpendicular  let 
fall  from  the  focus  on  the  tangent  of  the  section.  This  appears  from 
Cor.  5. 

COR.  9.  Wherefore  since  (by  Cor.  6,  Prop.  IV),  the  velocity  of  a  body 
revolving  in  this  circle  is  to  the  velocity  of  another  body  revolving  in  any 
other  circle  reciprocally  in  the  subduplicate  ratio  of  the  distances;  there 
fore,  ex  czqiiO)  the  velocity  of  a  body  revolving  *in  a  conic  section  will  be 
to  the  velocity  of  a  body  revolving*  in  a  circle  at  the  same  distance  as  a 
mean  proportional  between  that  common  distance,  and  half  the  principal 
latus  rectum  of  the  section,  to  the  perpendicular  let  fall  from  the  common 
focus  upon  the  tangent  of  the  section. 

PROPOSITION  XVII.     PROBLEM   IX. 

Supposing  the  centripetal  force  to  be  reciprocally  proportional  to  the 
squares  of  the  distances  of  places  from  the  centre,  and  that  the  abso 
lute  quantity  of  that  force  is  known  ;  it  is  required  to  determine  t/te 
line  which  a  body  will  describe  that  is  let  go  from  a  given  place  with  a 
given  velocity  in  the  direction  of  a  given  right  line. 
Let  the  centripetal  force 
tending  to  the  point  S  be 
such  as  will  make  the  body 
p  revolve  in  any  given  orbit 
pq ;  and  suppose  the  velocity 
of  this  body  in  the  place  p 
is  known.     Then  from  the 
place  P  suppose  the  body  P 
to  be  let  go  with  a  given  ve 
locity  in  the  direction  of  the 
line  PR ;  but  by  virtue  of  a 
centripetal  force  to  be  immediately  turned  aside  from  that  right  line  into 
the  conic  section  PQ,.     This,  the  right  line  PR  will  therefore  touch  in  P. 
Suppose  likewise  that  the  right  line  pr  touches  the  orbit  pq  in  p ;  and  if 
from  S  you  suppose  perpendiculars  let  fall  on  those  tangents,  the  principal 
latus  rectum  of  the  conic  section  (by  Cor.  1,  Prop.  XVI)  will  be  to  the 
principal  latus  rectum  of  that  orbit  in  a  ratio  compounded  of  the  duplicate 
ratio  of  the  perpendiculars,  and  the  duplicate  ratio  of  the  velocities ;  arid 
is  therefore  given.     Let  this  latus  rectum  be  L  ;  the  focus  S  of  the  conic 


L24  THE    MATHEMATICAL    PRINCIPLES  [BOOK    I 

section  is  also  given.  Let  the  angle  RPH  be  the  complement  of  the  angle 
RPS  to  two  right ;  and  the  line  PH,  in  which  the  other  focus  II  is  placed, 
is  given  by  position.  Let  fall  SK  perpendicular  on  PH,  and  erect  the 
conjugate  semi-axis  BC  ;  this  done,  we  shall  have  SP2—  2KPH  +  PH2 
=  SH2  =  4CH2  =  4BH2  — 4BC2  =  SP  +  PH2— L  X  SiM 


SP2  +  2SPH  +  PH2  —  L  x  SP  +  PH.     Add  on  both  sides  2KPH  — 


SP2— PH2  +  L  X  SP  +  PH,  and  we  shall  have  L  X  SP  +  PH  =  2SPH 
f  2KPH,  or  SP  +  PH  to  PH,  as  2SP  +  2KP  to  L.  Whence  PH  is 
given  both  in  length  and  position.  That  is,  if  the  velocity  of  the  body 
in  P  is  such  that  the  latus  rectum  L  is  less  than  2SP  +  2KP,  PH  will 
lie  on  the  same  side  of  the  tangent  PR  writh  the  line  SP ;  and  therefore 
the  figure  will  be  an  ellipsis,  which  from  the  given  foci  S,  H,  and  the 
principal  axis  SP  +  PH,  is  given  also.  But  if  the  velocity  of  the  body 
is  so  great,  that  the  latus  rectum  L  becomes  equal  to  2SP  +  2KP,  the 
length  PH  will  be  infinite ;  and  therefore,  the  figure  will  be  a  parabola, 
which  has  its  axis  SH  parallel  to  the  line  PK,  and  is  thence  given.  But 
if  the  body  goes  from  its  place  P  with  a  yet  greater  velocity,  the  length 
PH  is  to  be  taken  on  the  other  side  the  tangent ;  and  so  the  tangent  pas 
sing  between  the  foci,  the  figure  will  be  an  hyperbola  having  its  principal 
axis  equal  to  the  difference  of  the  lines  SP  and  PH,  and  thence  is  given. 
Por  if  the  body,  in  these  cases,  revolves  in  a  conic  section  so  found,  it  is 
demonstrated  in  Prop.  XI,  XII,  and  XIII,  that  the  centripetal  force  will 
be  reciprocally  as  the  square  of  the  distance  of  the  body  from  the  centre 
of  force  S ;  and  therefore  we  have  rightly  determined  the  line  PQ,,  which 
a  body  let  go  from  a  given  place  P  with  a  given  velocity,  and  in  the  di 
rection  of  the  right  line  PR  given  by  position,  would  describe  with  such  a 
force.  Q.E.F. 

COR.  1.  Hence  in  every  conic  section,  from  the  principal  vertex  D,  the 
latus  rectum  L,  and  the  focus  S  given,  the  other  focus  H  is  given,  by 
taking  DH  to  DS  as  the  latus  rectum  to  the  difference  between  the  latus 
rectum  and  4US.  For  the  proportion,  SP  +  PH  to  PH  as  2SP  +  2KP 
to  L,  becomes,  in  the  case  of  this  Corollary,  DS  +  DH  to  DH  as  4DS  to 
L,  and  by  division  DS  to  DH  as  4DS  —  L  to  L. 

COR.  2.  Whence  if  the  velocity  of  a  body  in  the  principal  vertex  D  ig 
given,  the  orbit  may  be  readily  found ;  to  wit,  by  taking  its  latus  rectum 
to  twice  the  distance  DS,  in  the  duplicate  ratio  of  this  given  velocity  to 
the  velocity  of  a  body  revolving  in  a  circle  at  the  distance  DS  (by  Cor. 
3,  Prop.  XVI.),  and  then  taking  DH  to  DS  as  the  latus  rectum  to  the 
difference  between  the  latus  rectum  and  4DS. 

COR.  3.  Hence  also  if  a  body  move  in  any  conic  section,  and  is  forced 
out  of  its  orbit  by  any  impulse,  you  may  discover  the  orbit  in  which  it  will 
afterwards  pursue  its  Bourse.  For  bv  compounding  the  proper  motion  oi 


SEC.    IV.]  OF    NATURAL    PHILOSOPHY.  125 

the  body  with  that  motion,  which  the  impulse  alone  would  generate,  you 
will  have  the  motion  with  which  the  body  will  go  off  from  a  given  place 
of  impulse  in  the  direction  of  a  right  line  given  in  position. 

COR.  4.  And  if  that  body  is  continually  disturbed  by  the  action  of  some 
foreign  force,  we  may  nearly  know  its  course,  by  collecting  the  changes 
which  that  force  introduces  in  some  points,  and  estimating  the  continual 
changes  it  will  undergo  in  the  intermediate  places,  from  the  analogy  that 
appears  in  the  progress  of  the  series. 

SCHOLIUM. 

If  a  body  P,  by  means  of  a  centripetal 
force  tending  to  any  given  point  R,  move 
in  the  perimeter  of  any  given  conic  sec 
tion  whose  centre  is  C  ;  and  the  law  of 
the  centripetal  force  is  required :  draw 
CG  parallel  to  the  radius  RP,  and  meet 
ing  the  tangent  PG  of  the  orbit  in  G ; 
and  the  force  required  (by  Cor.  1,  and 

CG3 
Schol.  Prop.  X.,  and  Cor.  3,  Prop.  VII.)  will  be  as     - 


SECTION  IV. 

Of  the  finding  of  elliptic,  parabolic,  and  hyperbolic  orbits,  from  ttu. 
focus  given. 

LEMMA  XV. 

If  from  the  two  foci  S,  II,  of  any  ellipsis  or  hyberbola,  we  draw  to  any 
third  point  V  the  right  lines  SV,  H V,  whereof  one  HV  is  equal  to  the 
principal  axis  of  the  figure,  thai  is,  to  the  axis  in  which  the  foci  are 
situated,  the  other,  SV,  is  bisected  in  T  by  t/ie  perpendicular  TR  let 
fall  upon  it ;  that  perpendicular  TR  will  somewhere  touch  the  conic 
section :  and,  vice  versa,  if  it  does  touch  it,  HV  will  be  equal  to  the 
principal  axis  of  the  figure. 
For,  let  the  perpendicular  TR  cut  the  right  line 
HV,  produced,  if  need  be,  in  R ;  and  join  SR.     Be 
cause  TS,  TV  are  equal,  therefore  the  right  lines  SR, 
VR,  as  well  as  the  angles  TRS,  TRV,  will  be  also 
equal.     Whence  the  point  R  will  be  in  the  conic  section,  and  the  perpen 
dicular  TR  will  touch  the  same ;  and  the  contrary.     Q.E.D. 


126  THE    MATHEMATICAL    PBINCIP,  -ES  [BOOK  1 

PROPOSITION  XVIII.     PROBLEM  X. 

From  a  focus  and  the  principal  axes  given,  to  describe  elliptic  and  hy 
perbolic  trajectories,  which  shall  pass  through  given  points,  and  touch 
right  lines  given  by  position. 
Let  S  be  the  common  focus  of  the  figures ;  AB  A 33 

the  length  of  the  principal  axis  of  any  trajectory ;     r          p  T~* 

P  a  point  through  which  the  trajectory  should  \    /R 

pass ;  and  TR  a  right  line  which  it  should  touch.  /  \ 


About  the  centre  P,  with  the  interval  AB  —  SP,  £\    S  ~~yf 

if  the  orbit  is  an  ellipsis,  or  AB    {-  SP,  if  the       y>  G   ^ 

orbit  is  an  hyperbola,  describe  the  circle  HG.  On  the  tangent  TR  let  fall 
the  perpendicular  ST,  and  produce  the  same  to  V,  so  that  TV  may  be 
equal  to  ST;  and  about  V  as  a  centre  with  the  interval  AB  describe  the 
circle  FH.  In  this  manner,  whether  two  points  P,  p,  are  given,  or  two 
tangents  TR,  tr,  or  a  point  P  and  a  tangent  TR,  we  are  to  describe  two 
circles.  Let  H  be  their  common  intersection,  and  from  the  foci  S,  H,  with 
the  given  axis  describe  the  trajectory  :  I  say,  the  thing  is  done.  For  (be 
cause  PH  -f-  SP  in  the  ellipsis,  and  PH  —  SP  in  the  hyperbola,  is  equal 
to  the  axis)  the  described  trajectory  will  pass  through  the  point  P,  and  (by 
the  preceding  Lemma)  will  touch  the  right  line  TR.  And  by  the  same 
argument  it  will  either  pass  through  the  two  points  P,  p,  or  touch  the  two 
right  lines  TR,  tr.  Q.E.F. 

PROPOSITION  XIX.     PROBLEM  XI. 

About  a  given  focus,  to  describe  a  parabolic  trajectory,  which  shall  pass 
through  given  points,  and  touch  right  lines  given  by  position. 
Let  S  be  the  focus,  P  a  point,  and  TR  a  tangent  of 
the  trajectory  to  be  described.  About  P  as  a  centre, 
with  the  interval  PS,  describe  the  circle  FG.  From 
the  focus  let  fall  ST  perpendicular  on  the  tangent,  and 
produce  the  same  to  V,  so  as  TV  may  be  equal  to  ST. 
After  the  same  manner  another  circle  fg  is  to  be  de 
scribed,  if  another  point  p  is  given  ;  or  another  point  v 
is  to  be  found,  if  another  tangent  tr  is  given;  then  draw 
the  right  line  IF,  which  shall  touch  the  two  circles  YG,fg,  if  two  points 
P,  p  are  given ;  or  pass  through  the  two  points  V,  v,  if  two  tangents  TR, 
tr,  are  given :  or  touch  the  circle  FG,  and  pass  through  the  point  V,  if  the 
point  P  and  the  tangent  TR  are  given.  On  FI  let  fall  the  perpendicular 
SI,  and  bisect  the  same  in  K ;  and  with  the  axis  SK  and  principal  vertex  K 
describe  a  parabola  :  I  say  the  thing  is  done.  For  this  parabola  (because 
SK  is  equal  to  IK,  and  SP  to  FP)  will  pass  through  the  point  P ;  and 


/KS 


SEC.  IV.]  OF    NATURAL    PHILOSOPHY.  127 

(by  Cor.  3,  Lem.  XIV)  because  ST  is  equal  to  TV.  and  STR  a  light  an 
gle,  it  will  touch  the  right  line  TR.  Q.E.F. 

PROPOSITION  XX.    PROBLEM  XII. 

About  a  given  focus  to  describe  any  trajectory  given  in  specie  which  shah 
pass  through  given  points,  and  touch  right  lines  given  by  position. 
CASE  1.  About  the  focus  S   it   is   re- 
uired  to  describe  a  trajectory  ABC,  pass 
ing  through  two  points  B,  C.     Because  the 
trajectory  is  given  in  specie,  the  ratio  of  the 

principal  axis  to  the  distance  of  the  foci     GAS  H 

will  be  given.  In  that  ratio  take  KB  to  BS,  and  LC  to  CS.  About  the 
centres  B,  C,  with  the  intervals  BK,  CL,  describe  two  circles ;  and  on  the 
right  line  KL,  that  touches  the  same  in  K  and  L,  let  fall  the  perpendicu 
lar  SG ;  which  cut  in  A  and  a,  so  that  GA  may  be  to  AS,  and  Ga  to  aS, 
as  KB  to  BS  ;  and  with  the  axis  A.«,  and  vertices  A,  a,  describe  a  trajectory  : 
I  say  the  thing  is  done.  For  let  H  be  the  other  focus  of  the  described 
figure,  and  seeing  GA  is  to  AS  as  Ga  to  aS,  then  by  division  we  shall 
have  Ga — GA,  or  Aa  to  «S — AS,  or  SH  in  the  same  ratio;  and  therefore 
in  the  ratio  which  the  principal  axis  of  the  figure  to  be  described  has  to 
the  distance  of  its  foci ;  and  therefore  the  described  figure  is  of  the  same 
species  with  the  figure  which  was  to  be  described.  And  since  KB  to  BS, 
and  LC  to  CS,  are  in  the  same  ratio,  this  figure  will  pass  through  tht- 
points  B,  C,  as  is  manifest  from  the  conic  sections. 

CASE  2.  About  the  focus  S  it  is  required  to 
describe  a  trajectory  which  shall  somewhere 
touch  two  right  lines  TR,  tr.  From  the  focus 
on  those  tangents  let  fall  the  perpendiculars 
ST,  St,  which  produce  to  V,  v,  so  that  TV,  tv 
may  be  equal  to  TS,  tS.  Bisect  Vv  in  O,  and  j 
erect  the  indefinite  perpendicular  OH,  and  cut  I. 
the  right  line  VS  infinitely  produced  in  K  and  V 
k,  so  that  VK  be  to  KS,  and  VA*  to  A~S,  as  the  principal  axis  of  the  tra 
jectory  to  be  described  is  to  the  distance  of  its  foci.  On  the  diameter 
K/J  describe  a  circle  cutting  OH  in  H ;  and  with  the  foci  S,  H,  and 
principal  axis  equal  to  VH,  describe  a  trajectory  :  I  say,  the  thing  is  done. 
For  bisecting  Kk  in  X,  and  joining  HX,  HS,  HV,  Hv,  because  VK  is  to 
KS  as  VA-  to  A*S ;  and  by  composition,  as  VK  -f-  V/c  to  KS  +  kS ;  and 
by  division,  as  VA*  —  VK  to  kS  —  KS,  that  is,  as  2VX  to  2KX,  and 
2KX  to  2SX,  and  therefore  as  VX  to  HX  and  HX  to  SX,  the  triangles 
VXH,  HXS  will  be  similar ;  therefore  VH  will  be  to  SH  as  VX  to  XH ; 
and  therefore  as  VK  to  KS.  Wherefore  VH,  the  principal  axis  of  the 
described  trajectory,  has  the  same  ratio  to  SH,  the  distance  of  the  foci,  as 


12S 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK    1. 


K     S 


the  principal  axis  of  the  trajectory  which  was  to  be  described  has  to  the 
distance  of  its  foci ;  and  is  therefore  of  the  same  species.  Arid  seeing  VH, 
vH  are  equal  to  the  principal  axis,  and  VS,  vS  are  perpendicularly  bisected 
by  the  right  lines  TR,  tr,  it  is  evident  (by  Lem.  XV)  that  those  right 
lines  touch  the  described  trajectory.  Q,.E.F. 

CASE.  3.  About  the  focus  S  it  is  required  to  describe  a  trajectory,  which 
shall  touch  a  right  line  TR  in  a  given  Point  R.  On  the  right  line  TR 
Jet  fall  the  perpendicular  ST,  which  produce  to  V,  so  that  TV  may  be 
equal  to  ST ;  join  VR,  and  cut  the  right  line  VS  indefinitely  produced 
in  K  and  k,  so.  that  VK  may  be  to  SK,  and  V&  to  SAr,  as  the  principal 
axis  of  the  ellipsis  to  be  described  to  the  distance  of  its  foci ;  and  on  the 
diameter  KA:  describing  a  circle,  cut  the  H 

right  line  VR  produced  in  H ;  then  with 
the  foci  S,  H,  and  principal  axis  equal  to  R 
VH,  describe  a  trajectory  :  I  say,  the  thing  .---' 
is  done.  For  VH  is  to  SH  as  VK  to  SK,  V"  "1 
and  therefore  as  the  principal  axis  of  the  trajectory  which  was  to  be  de 
scribed  to  the  distance  of  its  foci  (as  appears  from  what  we  have  demon 
strated  in  Case  2) ;  and  therefore  the  described  trajectory  is  of  the  same 
species  with  that  which  was  to  be  described ;  but  that  the  right  line  TR, 
by  which  the  angle  VRS  is  bisected,  touches  the  trajectory  in  the  point  R, 
is  certain  from  the  properties  of  the  conic  sections.  Q.E.F. 

CASE  4.  About  the  focus  S  it  is  r 

required    to    describe   a  trajectory 
APB  that  shall  touch  a  right  line 
TR,  and  pass  through  any  given 
point  P  without  the  tangent,  and 
shall  be  similar  to  the  figure  apb, 
described  with  the  principal  axis  ab, 
and  foci  s,  h.     On  the  tangent  TR 
let  fall  the  perpendicular  ST,  which         /    ..''''',.---"'" 
produce  to  V,  so  that  TV  may  be 
equal  to  ST ;  and  making  the  an 
gles  hsq,  shq,  equal  to  the  angles  VSP,  SVP,  about  q  as  a  centre,  and 
with  an  interval  which  shall  be  to  ab  as  SP  to  VS,  describe  a  circle  cut 
ting  the  figure  apb  in  p :  join  sp,  and  draw 
SH  such  that  it  may  be  to  sh  as  SP  is  to  sp, 
and  may  make  the  angle  PSH  equal  to  the 
angle  psh,  and  the  angle  VSH  equal  to  the 
angle  pyq.     Then  with  the  foci  S,  H,  and  B 
principal  axis  AB,  equal  to  the  distance  VH, 
describe  a  conic  section :   I  say,  the  thing  is 
done ;  for  if  sv  is  drawn  so  that  it  shall  be  to 


SEC.  IV.]  OF    NATURAL    PHILOSOPHY.  129 

sp  as  sh  is  to  sq,  and  shall  make  the  angle  vsp  equal  to  the  angle  hsq,  and 
the  angle  vsh  equal  to  the  angle  psq,  the  triangles  svh,  spq,  will  be  similar, 
and  therefore  vh  will  be  to  pq  as  sh  is  to  sq  ;  that  is  (because  of  the  simi 
lar  triangles  VSP,  hsq),  as  VS  is  to  SP?  or  as  ab  to  pq.  Wherefore 
vh  and  ab  are  equal.  But,  because  of  the  similar  triangles  VSH,  vsh,  VH 
is  to  SH  as  vh  to  sh ;  that  is,  the  axis  of  the  conic  section  now  described 
is  to  the  distance  of  its  foci  as  the  axis  ab  to  the  distance  of  the  foci  sh  ; 
and  therefore  the  figure  now  described  is  similar  to  the  figure  aph.  But, 
because  the  triangle  PSH  is  similar  to  the  triangle  psh,  this  figure  passes 
through  the  point  P ;  and  because  VH  is  equal  to  its  axis,  and  VS  is  per 
pendicularly  bisected  by  the  rght  line  TR,  the  said  figure  touches  the 
right  line  TR.  Q.E.F. 

LEMMA  XVI. 

From  three  given  points  to  draw  to  afonrth  point  that  is  not  given  three 
right  lines  whose  differences  shall  be  either  given,  or  none  at  all. 
CASE  1.  Let  the  given  points  be  A,  B,  C,  and  Z  the  fourth  point  which 
we  are  to  find ;  because  of  the  given  difference  of  the  lines  AZ,  BZ,  the 
locus  of  the  point  Z  will  be  an  hyperbola 
whose  foci  are  A  and  B,  and  whose  princi 
pal  axis  is  the  given  difference.  Let  that 
axis  be  MN.  Taking  PM  to  MA  as  MN 
is  to  AB,  erect  PR  perpendicular  to  AB, 
and  let  fall  ZR  perpendicular  to  PR ;  then 
from  the  nature  of  the  hyperbola,  ZR  will 
be  to  AZ  as  MN  is  to  AB.  And  by  the 
like  argument,  the  locus  of  the  point  Z  will 
be  another  hyperbola,  whose  foci  are  A,  C,  and  whose  principal  axis  is  the 
difference  between  AZ  and  CZ ;  and  QS  a  perpendicular  on  AC  may  be 
drawn,  to  which  (QS)  if  from  any  point  Z  of  this  hyperbola  a  perpendicular 
ZS  is  let  fall  (this  ZS),  shall  be  to  AZ  as  the  difference  between  AZ  and 
CZ  is  to  AC.  Wherefore  the  ratios  of  ZR  and  ZS  to  AZ  are  given,  and 
consequently  the  ratio  of  ZR  to  ZS  one  to  the  other  ;  and  therefore  if  the 
right  lines  RP,  SQ,  meet  in  T,  and  TZ  and  TA  are  drawn,  the  figure 
TRZS  will  be  given  in  specie,  and  the  right  line  TZ,  in  which  the  point 
Z  is  somewhere  placed,  will  be  given  in  position.  There  will  be  given 
also  the  right  line  TA,  and  the  angle  ATZ ;  and  because  the  ratios  of  AZ 
and  TZ  to  ZS  are  given,  their  ratio  to  each  other  is  given  also ;  and 
thence  will  be  given  likewise  the  triangle  ATZ,  whose  vertex  is  the  point 
Z.  Q.E.I. 

CASE  2.  If  two  of  the  three  lines,  for  example  AZ  and  BZ,  are  equal, 
draw  the  right  line  TZ  so  as  to  bisect  the  right  line  AB ;  then  find  the 
triangle  ATZ  as  above.  Q.E.I. 


130 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK  I. 


CASE  3.  If  all  the  three  are  equal,  the  point  Z  will  be  placed  in  the 
centre  of  a  circle  that  passes  through  the  points  A,  B,  C.  Q.E.I. 

This  problematic  Lemma  is  likewise  solved  in  Apollonius's  Book  oi 
Tactions  restored  by  Vieta. 

PROPOSITION  XXL     PROBLEM   XIII. 

About  a  given  focus  to  describe  a  trajectory  that  shall  pass  through 
given  points  and  touch  right  Hues  given  by  position. 
Let  the  focus  S,  the  point  P,  and  the  tangent  TR  be  given,  and  suppose 

that  the  other  focus  H  is  to  be  found. 

On  the  tangent  let  fall  the  perpendicular 

ST,  which  produce  to  Y,  so  that  TY  may 

be  equal  to  ST,  and  YH  will  be  equal 

to  the  principal  axis.     Join  SP,  HP,  and 

SP  will  be  the  difference  between  HP  and 

the  principal  axis.     After  this  manner, 

if  more  tangents  TR  are  given,  or  more 

points  P.  we  shall  always  determine  as 

many  lines  YH,  or  PH,  drawn  from  the  said  points  Y  or  P,  to  the  focus 

H,  which  either  shall  be  equal  to  the  axes,  or  differ  from  the  axes  by  given 

lengths  SP  ;  and  therefore  which  shall  either  be  equal  among  themselves, 

or  shall  have  given  differences ;  from  whence  (by  the  preceding  Lemma). 

that  other  focus  H  is  given.     But  having  the  foci  and  the  length  of  the 

axis  (which  is  either  YH,  or,  if  the  trajectory  be  an  ellipsis,  PH  -f  SP ; 

or  PH  —  SP,  if  it  be  an  hyperbola),  the  trajectory  is  given.     Q.E.I. 

SCHOLIUM. 

When  the  trajectory  is  an  hyperbola,  I  do  not  comprehend  its  conjugate 
hyperbola  under  the  name  of  tins  trajectory.  For  a  body  going  on  with  a 
continued  motion  can  never  pass  out  of  one  hyperbola  into  its  conjugate 
hyperbola. 

The  case  when  three  points  are  given 
is  more  readily  solved  thus.  Let  B,  C, 
I),  be  the  given  points.  Join  BC,  CD, 
and  produce  them  to  E,  F,  so  as  EB  may 
be  to  EC  as  SB  to  SC ;  and  FC  to  FD 
as  SC  to  SD.  On  EF  drawn  and  pro 
duced  let  fall  the  perpendiculars  SG, 
BH,  and  in  GS  produced  indefinitely  E 
take  GA  to  AS,  and  Ga  to  aS,  as  HB 
is  to  BS ;  then  A  will  be  the  vertex,  and  Aa  the  principal  axis  of  the  tra 
jectory  ;  which,  according  as  GA  is  greater  than,  equal  to,  or  less  than 


SEC.  V.]  OF    NATURAL    PHILOSOPHY.  131 

AS.  will  be  either  an  ellipsis,  a  parabola,  or  an  hyperbola ;  the  point  a  in 
the  first  case  falling  on  the  same  side  of  the  line  GP  as  the  point  A  ;  in 
the  second,  going  oft*  to  an  infinite  distance ;  in  the  third,  falling  on  the 
other  side  of  the  line  GP.  For  if  on  GF  the  perpendiculars  CI,  DK  are 
let  fall,  TC  will  be  to  HB  as  EC  to  EB ;  that  is,  as  SO  to  SB ;  and  by 
permutation,  1C  to  SC  as  HB  to  SB,  or  as  GA  to  SA.  And,  by  the  like 
argument,  we  may  prove  that  KD  is  to  SD  in  the  same  ratio.  Where 
fore  the  points  B,  C,  D  lie  in  a  conic  section  described  about  the  focus  S, 
in  such  manner  that  all  the  right  lines  drawn  from  the  focus  S  to  the 
several  points  of  the  section,  and  the  perpendiculars  let  fall  from  the  same 
points  on  the  right  line  GF,  are  in  that  given  ratio. 

That  excellent  geometer  M.  De  la  Hire  has  solved  this  Problem  much 
after  the  same  way,  in  his  Conies,  Prop.  XXV.,  Lib.  VIII. 


SECTION  V. 

How  the  orbits  are  to  be  found  when  neither  focus  is  given. 

LEMMA  XVII. 

If  from  any  point  P  of  a  given  conic  section,  to  the  four  produced  sides 
AB,  CD,  AC,  DB,  of  any  trapezium  ABDC  inscribed  in  that  section, 
as  many  right  lines  PQ,  PR,  PS,  PT  are  drawn  in  given  ang7ei, 
each  line  to  each  side  ;  the  rectangle  PQ,  X  PR  of  those  on  the  opposite 
sides  AB,  CD,  will  be  to  the  rectangle  PS  X  PT  of  those  on  tie  other 
two  opposite  sides  AC,  BD,  in  a  given  ratio. 
CASE  1.  Let  us  suppose,  first,  that  the  lines  drawn 

to  one  pair  of  opposite  sides  are  parallel  to  either  of     I  ^^        p       ;T 
the  other  sides ;  as  PQ  and  PR  to  the  side  AC,  and  s| 
PS  and  PT  to  the  side  AB.     And  farther,  that  one 
pair  of  the  opposite  sides,  as  AC  and  BD,  are  parallel 
betwixt  themselves;  then  the  right  line  which  bisects^  IQ      I3 

those  parallel  sides  will  be  one  of  the  diameters  of  the  1L 

conic  section,  and  will  likewise  bisect  RQ.  Let  O  be  the  point  in  which 
RQ  is  bisected,  and  PO  will  be  an  ordinate  to  that  diameter.  Produce 
PO  to  K,  so  that  OK  may  be  equal  to  PO,  and  OK  will  be  an  ordinate 
on  the  other  side  of  that  diameter.  Since,  therefore,  the  points  A,  B;  P 
and  K  are  placed  in  the  conic  section,  and  PK  cuts  AB  in  a  given  angle, 
the  rectangle  PQK  (by  Prop.  XVII.,  XIX.,  XXI.  and  XXI1L,  Book  III., 
of  Apollonius's  Conies)  will  be  to  the  rectangle  AQB  in  a  given  ratio. 
But  QK  and  PR  are  equal,  as  being  the  differences  of  the  equal  lines  OK, 
OP,  and  OQ,  OR  ;  whence  the  rectangles  PQK  and  PQ  X  PR  are  equal ; 
and  therefore  the  rectangle  PQ  X  PR  is  to  the  rectangle  A^  B,  that  Is,  to 
the  rectangle  PS  X  PT  in  a  given  ratio.  Q.E.D 


132 


THE   MATHEMATICAL    PRINCIPLES 


[BOOK    I 


CASE  2.  Let  us  next  suppose  that  the  oppo 
site  sides  AC  and  BD  of  the  trapezium  are  not 
parallel.  Draw  Be/  parallel  to  AC,  and  meeting 
as  well  the  right  line  ST  in  /,  as  the  conic  section 
in  d.  Join  Cd  cutting  PQ  in  r,  and  draw  DM 
parallel  to  PQ,  cutting  Cd  in  M,  and  AB  in  N. 

Then  (because  of  the  similar  triangles  BTt, 

DBN),  Et  or  PQ  is  to  Tt  as  DN  to  NB.     And  ^^  Q       N 

so  Rr  is  to  AQ  or  PS  as  DM  to  AN.  Wherefore,  by  multiplying  the  antece- 
dents  by  the  antecedents,  and  the  consequents  by  the  consequents,  as  the 
rectangle  PQ  X  Rr  is  to  the  rectangle  PS  X  Tt,  so  will  the  rectangle 
N  i)M  be  to  the  rectangle  ANB ;  and  (by  Case  1)  so  is  the  rectangle 
PQ  X  Pr  to  the  rectangle  PS  X  Pt :  and  by  division,  so  is  the  rectangle 
PQ  X  PR  to  the  rectangle  PS  X  PT.  Q.E.D. 

CASE  3.  Let  us  suppose,  lastly,  the  four  lines 
?Q,  PR,  PS,  PT,  not  to  be  parallel  to  the  sides 
AC,  AB,  but  any  way  inclined  to  them.     In  their 
place  draw  Pq,  Pr,  parallel  to  AC  ;  and  Ps,  Pt 
parallel  to   AB ;  and  because  the  angles   of  the 
triangles  PQ</,  PRr,  PSs,  PTt  are  given,  the  ra- 
tios  of  IQ  to  Pq,  PR  to  Pr,  PS  to  P*,  PT  to  Pt 
will  b?  also  given;  and  therefore  the  compound 
ed  ratios  Pk    X   PR  to  P?   X    Pr,  and  PS    X    PT   to   Ps   X    Pt  are 
given.     But  from  what  we  have  demonstrated  before,  the  ratio  of  Pq  X  Pi 
to  Ps  X  Pt  is  given ;  and  therefore  also  the  ratio  of  PQ  X  PR  to  PS  X 
PT.     Q.E.D. 

LEMMA  XVIII. 

The  s  'niL  things  supposed,  if  the  rectangle  PQ  X  PR  of  the  lines  drawn 
to  the  two  opposite  sides  of  the  trapezium  is  to  the  rectangle  PS  X  PT 
of  those  drawn  to  the  other  two  sides  in  a  given  ratio,  the  point  P, 
from  whence  those  lines  are  drawn,  will  be  placed  in  a  conic  section 
described  about  the  trapezium. 
Conceive  a  conic  section  to  be  described  pas 
sing  through  the  points  A,  B,  C,  D,  and  any 

one  of  the  infinite  number  of  points  P,  as  for 

example  p ;  I  say,  the  point  P  will  be  always  c1 

placed  in  this  section.     If  you  deny  the  thing, 

join  AP  cutting  this  conic  section  somewhere 

else,  if  possible,  than  in  P,  as  in  b.     Therefore 

if  from  those  points  p  and  b,  in  the  given  angles      ^  B 

to  the  sides  of  the  trapezium,  we  draw  the  right 

lines pq,  pr,  ps,  pt,  and  bk,  bn,  bf,  bd,  we  shall  have,  as  bk  X  bn  to  bf  X  bd, 


SEC.  V.]  OF    NATURAL    PHILOSOPHY  133 

so  (by  Lem.  XVII)  pq  X  pr  to  ps  X  pt ;  and  so  (by  supposition)  PQ  x 
PR  to  PS  X  PT.  And  because  of  the  similar  trapezia  bkAf,  PQAS,  as 
bk  to  bf,  so  PQ  to  PS.  Wherefore  by  dividing  the  terms  of  the  preceding 
proportion  by  the  correspondent  terms  of  this,  we  shall  have  bn  to  bd  as 
PR  to  PT.  And  therefore  the  equiangular  trapezia  ~Dnbd,  DRPT,  are 
similar,  and  consequently  their  diagonals  D6,  DP  do  coincide.  Wherefore 
b  falls  in  the  intersection  of  the  right  lines  AP,  DP,  and  consequently 
coincides  with  the  point  P.  And  therefore  the  point  P,  wherever  it  is 
taken,  falls  to  be  in  the  assigned  conic  section.  Q.E.D. 

COR.  Hence  if  three  right  lines  PQ,  PR,  PS,  are  drawn  from  a  com 
mon  point  P,  to  as  many  other  right  lines  given  in  position,  AB,  CD,  AC, 
each  to  each,  in  as  many  angles  respectively  given,  and  the  rectangle  PQ 
X  PR  under  any  two  of  the  lines  drawn  be  to  the  square  of  the  third  PS 
in  a  given  ratio ;  the  point  P,  from  which  the  right  lines  are  drawn,  will 
be  placed  in  a  conic  section  that  touches  the  lines  AB;  CD  in  A  and  C  • 
and  the  contrary.  For  the  position  of  the  three  right  lines  AB,  CD,  AC 
remaining  the  same,  let  the  line  BD  approach  to  and  coincide  with  the 
line  AC ;  then  let  the  line  PT  come  likewise  to  coincide  with  the  line  PS ; 
and  the  rectangle  PS  X  PT  will  become  PS2,  and  the  right  lines  AB,  CD, 
which  before  did  cut  the  curve  in  the  points  A  and  B,  C  and  D,  can  no 
(onger  cut,  but  only  touch,  the  curve  in  those  coinciding  points. 

SCHOLIUM. 

In  this  Lemma,  the  name  of  conic  section  is  to  be  understood  in  a  large 
sense,  comprehending  as  well  the  rectilinear  section  through  the  vertex  of 
the  cone,  as  the  circular  one  parallel  to  the  base.  For  if  the  point  p  hap 
pens  to  be  in  a  right  line,  by  which  the  points  A  and  D,  or  C  and  B  are 
joined,  the  conic  section  will  be  changed  into  two  right  lines,  one  of  which 
is  that  right  line  upon  which  the  point  p  falls, 
and  the  other  is  a  right  line  that  joins  the  other 
two  of  *he  four  points.  If  the  two  opposite  an 
gles  of  the  trapezium  taken  together  are  equal  c 
to  two  right  angles,  and  if  the  four  lines  PQ, 
PR,  PS,  PT,  are  drawn  to  the  sides  thereof  at 
right  angles,  or  any  other  equal  angles,  and  the 
rectangle  PQ  X  PR  under  two  of  the  lines 
drawn  PQ  and  PR,  is  equal  to  the  rectangle 
PS  X  PT  under  the  other  two  PS  and  PT,  the  conic  section  will  become 
a  circle.  And  the  same  thing  will  happen  if  the  four  lines  are  drawn  in 
any  angles,  and  the  rectangle  PQ  X  PR,  under  one  pair  of  the  lines  drawn, 
is  to  the  rectangle  PS  X  PT  under  the  other  pair  as  the  rectangle  under 
the  sines  of  the  angles  S,  T,  in  which  the  two  last  lines  PS,  PT  are  drawn 
to  the  rectangle  under  the  sines  of  the  angles  Q,  R,  in  which  the  first  tw« 


134  THE    MATHEMATICAL    PRINCIPLES  [BOOK  1. 

PQ,  PR  are  drawn.  In  all  other  cases  the  locus  of  the  point  P  will  be 
one  of  the  three  figures  which  pass  commonly  by  the  name  of  the  conic 
sections.  But  in  room  of  the  trapezium  A  BCD,  we  may  substitute  a 
quadrilateral  figure  whose  two  opposite  sides  cross  one  another  like  diago 
nals.  And  one  or  two  of  the  four  points  A,  B,  C,  D  may  be  supposed  to 
be  removed  to  an  infinite  distance,  by  which  means  the  sides  of  the  figure 
which  converge  to  those  points,  will  become  parallel ;  and  in  this  case  the 
conic  section  will  pass  through  the  other  points,  and  will  go  the  same  way 
as  the  parallels  in,  infinitum. 

LEMMA  XIX. 

To  find  a  point  P  from  which  if  four  right  lines  PQ,  PR,  PS,  PT  an 
drawn  to  as  many  other  right  lines  AB,  CD,  AC,  BD,  given  by  posi 
tion,  each  to  each,  at  given  angles,  the  rectangle  PQ  X  PR,  under  any 
two  of  the  lines' drawn,  shall  be  to  the  rectangle  PS  X  PT,  under  the 
other  tivo.  in  a  given  ratio. 
Suppose  the  lines  AB,  CD,  to  which  the  two 
right  lines  PQ,  PR,  containing  one  of  the  rect 
angles,  are  drawn  to  meet  two  other  lines,  given 
by  position,  in  the  points  A,  B,  C,  D.     From  one 
of  those,  as  A,  draw  any  right  line  AH,  in  which 
you  would  find  the  point  P.     Let  this  cut  the 
opposite  lines  BD,  CD,  in  H  and  I ;  and,  because 
all  the  angles  of  the  figure  are  given,  the  ratio  of 
PQ  to  PA,  and  PA  to  PS,  and  therefore  of  PQ 

to  PS,  will  be  also  given.  Subducting  this  ratio  from  the  given  ratio  oi 
PQ  X  PR  to  PS  X  PT,  the  ratio  of  PR  to  PT  will  be  given ;  and  ad 
ding  the  given  ratios  of  PI  to  PR,  and  PT  to  PH,  the  ratio  of  PI  to  PH. 
and  therefore  the  point  P  will  be  given.  Q.E.I. 

COR.  1.  Hence  also  a  tangent  may  be  drawn  to  any  point  D  of  the 
locus  of  all  the  points  P.  For  the  chord  PD,  where  the  points  P  and  D 
meet,  that  is,  where  AH  is  drawn  through  the  point  D,  becomes  a  tangent. 
In  which  case  the  ultimate  ratio  of  the  evanescent  lines  IP  and  PH  will 
be  found  as  above.  Therefore  draw  CF  parallel  to  AD,  meeting  BD  in 
F,  and  cut  it  in  E  in  the  same  ultimate  ratio,  then  DE  will  be  the  tan 
gent  ;  because  CF  and  the  evanescent  IH  are  parallel,  and  similarly  cut  in 
E  and  P. 

COR.  2.  Hence  also  the  locus  of  all  the  points  P  may  be  determined. 
Through  any  of  the  points  A,  B,  C,  D,  as  A,  draw  AE  touching  the  locus, 
and  through  any  other  point  B  parallel  to  the  tangent,  draw  BF  meeting 
the  locus  in  F  ;  and  find  the  point  F  by  this  Lemma.  Bisect  BF  in  G, 
and,  drawing  the  indefinite  line  AG,  this  will  be  the  position  of  the  dia 
meter  to  which  BG  and  FG  are  ordinates.  Let  this  AG  meet  the  locus 


SEC.  V.J 


OF    NATURAL    PHILOSOPHY. 


in  H,  and  AH  will  be  its  diameter  or  latus  trans- 
versum.  to  which  the  latus  rectum  will  be  as  BG2 
to  AG  X  GH.  If  AG  nowhere  meets  the  locus, 
the  line  AH  being  infinite,  the  locus  will  be  a  par 
abola  ;  and  its  latus  rectum  corresponding  to  the 


diameter  AG  will  be  -.-7^ 
AC* 


But  if  it  does  meet  it 


anywhere,  the  locus  will  be  an  hyperbola,  when 

the  points  A  and  H  are  placed  on  the  same  side  the  point  G  ;  and  an 
ellipsis,  if  the  point  G  falls  between  the  points  A  and  H  ;  unless,  perhaps, 
the  angle  AGB  is  a  right  angle,  and  at  the  same  time  BG2  equal  to  the 
rectangle  AGH,  in  which  case  the  locus  will  be  a  circle. 

And  so  we  have  given  in  this  Corollary  a  solution  of  that  famous  Prob 
lem  of  the  ancients  concerning  four  lines,  begun  by  Euclid,  and  carried  on 
by  Apollonius  ;  and  this  not  an  analytical  calculus,  but  a  geometrical  com 
position,  such  as  the  ancients  required. 

LEMMA  XX. 

If  the  two  opposite  angular  points  A  and  P  of  any  parallelogram  ASPQ 
touch  any  conic  section  in  the  points  A  and  P  ;  and  the  sides  AQ,  AS 
of  one  of  those  angles,  indefinitely  produced,  meet  the  same  conic  section 
in  B  and  C  ;  and  from  the  points  of  concourse,  B  and  C  to  any  fifth 
point  D  of  the  conic  section,  two  right  lines  BD,  CD  are  drawn  meet- 
ing  tlie  two  other  sides  PS,  PQ  of  the  parallelogram,  indefinitely  pro 
duced  in  T  and  R  ;  the  parts  PR  and  PT,  cut  off  from  the  sides,  will 
always  be  one  to  the  other  in  a  given  ratio.  And  vice  versa,  if  those 
parts  cut  off'  are  one  to  the  other  in  a  given  ratio,  the  locus  of  the  point 
D  will  be  a  conic  section  passing  through  the  four  points  A,  B,  C,  F 
CASE  1.  Join  BP,  CP,  and  from  the  point 

D  draw  the  two  right  lines  DG,  DE,  of  which 

the  first  DG  shall  be  parallel  to  AB,  and 

meet  PB,  PQ,  CA  in  H,  I,  G  ;  and  the  other 

DE  shall  be  parallel  to  AC,  and  meet  PC, 

PS,  AB,  in  F,  K,  E  ;  and  (by  Lem.  XVII) 

the  rectangle  DE  X  DF  will  be  to  the  rect 

angle  DG    X    DH  in   a  given  ratio.     But 

PQ  is  to  DE  (or  IQ)  as  PB  to  HB,  and  con 

sequently  as  PT  to  DH  ;  and  by  permutation  PQ,  is  to  PT  as  DE  to 

DH.     Likewise  PR  is  to  DF  as  RC  to  DC,  and  therefore  as  (IG  or)  PS 

to  DG  ;  and  by  permutation  PR  is  to  PS  as  DF  to  DG  ;  and,  by  com 

pounding  those  ratios,  the  rectangle  PQ  X  PR  will  be  to  the  rectangle 

PS  X  PT  as  the  rectangle  DE  X  DF  is  to  the  rectangle  DG  X  DH. 

and  consequently  in  "a  given  ratio.     But  PQ  and  PS  are  given,  and  there 

fore  the  ratio  of  PR  to  PT  is  given.     Q.E.D. 


136 


THE    MATHEMATICAL    PRINCIPLES 


CASE  2.  But  if  PR  and  PT  are  supposed  to  be  in  a  given  ratio  one  to 
the  other,  then  by  going  back  again,  by  a  like  reasoning,  it  will  follow 
that  the  rectangle  DE  X  DF  is  to  the  rectangle  DG  X  DH  in  a  given 
rati) ;  and  so  the  point  D  (by  Lem.  XVIII)  will  lie  in  a  conic  section  pass 
ing  through  the  points  A.,  B,  C,  P,  as  its  locus.  Q.E.I). 

COR.  1.  Hence  if  we  draw  BC  cutting  PQ  in  r  and  in  PT  take  Pt  to 
Pr  in  the  same  ratio  which  PT  has  to  PR ;  then  Et  will  touch  the  conic 
section  in  the  point  B.  For  suppose  the  point  D  to  coalesce  with  the  point 
B,  so  that  the  chord  BD  vanishing,  BT  shall  become  a  tangent,  and  CD 
and  BT  will  coincide  with  CB  and  Bt. 

COR.  2.  And,  vice  versa,  if  Bt  is  a  tangent,  and  the  lines  BD,  CD  meet 
in  any  point  D  of  a  conic  section,  PR  will  be  to  PT  as  Pr  to  Pt.  And, 
on  the  contrary,  if  PR  is  to  PT  as  Pr  to  Pt,  then  BD  and  CD  will  meet 
in  some  point  D  of  a  conic  section. 

COR.  3.  One  conic  section  cannot  cut  another  conic  section  in  more  than 
four  points.  For,  if  it  is  possible,  let  two  conic  sections  pass  through  the 
h've  points  A,  B,  C,  P,  O ;  and  let  the  right  line  BD  cut  them  in  the 
points  D,  d,  and  the  right  line  Cd  cut  the  right  line  PQ,  in  q.  Therefore 
PR  is  to  PT  as  Pq  to  PT :  whence  PR  and  Pq  are  equal  one  to  the  other, 
against  the  supposition. 

LEMMA  XXI. 

If  two  moveable  and  indefinite  right  lines  BM,  CM  drawn  through  given 
points  B,  C,  as  poles,  do  by  their  point  of  concourse  M  describe  a  third 
right  line  MN  given  by  position  ;  and  other  two  indefinite  right  lines 
BD,CD  are  drawn,  making  with  the  former  two  at  those  given  points 
B,  C,  given  angles,  MBD,  MCD  :  I  say,  that  those  two  right  lines  BD, 
CD  will  by  their  point  of  concourse  D  describe  a  conic  section  passing 
through  the  points  B,  C.  And,  vice  versa,  if  the  right  lints  BD,  CD 
do  by  their  point  of  concourse  D  describe  a  conic  section  passing 
through  the  given  points  B,  C,  A,  and  the  angle  DBM  is  always 
equal  to  the  giren  angle  ABC,  as  well  as  the  angle  DCM  always 
equal  to  the  given  angle  ACB,  the  point  M  will  lie  in  a  right  line 
given  by  position,  as  its  locus. 
For  in  the  right  line  MN  let  a  point 

N  be  given,  and  when  the  moveable  point 

M  falls  on  the  immoveable  point  N.  let 

the  moveable  point  D  fall  on  an  immo 
vable  point  P.     Join   ON,  BN,  CP,  BP, 

and  from  the  point  P  draw  the  right  lines 

PT,  PR  meeting  BD,  CD  in  T  and  R,  C 

and  making  the  angle  BPT  c  jual  to  the 

given  angle  BNM,  and  the  angle  CPR 


SEC.  V.J 


OF    NATURAL    PHILOSOPHY. 


137 


equal  to  the  given  angle  CNM.  Wherefore  since  (by  supposition)  the  an 
gles  MBD,  NBP  are  equal,  as  also  the  angles  MOD,  NCP,  take  away  the 
angles  NBD  and  NOD  that  are  common,  and  there  will  remain  the  angles 
NBM  and  PBT,  NCM  and  PCR  equal;  and  therefore  the  triangles  NBM, 
PBT  are  similar,  as  also  the  triangles  NCM,  PCR.  Wherefore  PT  is  to 
NM  as  PB  to  NB  ;  and  PR  to  NM  as  PC  to  NC.  But  the  points,  B,  C, 
N,  P  are  immovable:  wheiefore  PT  and  PR  have  a  given  ratio  to  NM, 
and  consequently  a  given  ratio  between  themselves;  and  therefore,  (by 
Lemma  XX)  the  point  D  wherein  the  moveable  right  lines  BT  and  CR 
perpetually  concur,  will  be  placed  in  a  conic  section  passing  through  the 
points  B.  C,  P.  Q.E.D. 

And,  vice  versa,  if  the  moveable  point 
D  lies  in  a  conic  section  passing  through 
the  given  points  B,  C,  A  ;  and  the  angle 
DBM  is  always  equal  to  the  given  an 
gle  ABC,  and  the  angle  DCM  always 
equal  to  the  given  angle  ACB,  and  when 
the  point  D  falls  successively  on  any 
two  immovable  points  p,  P,  of  the  conic 
section,  the  moveable  point  M  falls  suc 
cessively  on  two  immovable  points  /?,  N. 
Through  these  points  ??,  N,  draw  the  right  line  nN :  this  line  nN  will  be 
the  perpetual  locus  of  that  moveable  point  M.  For,  if  possible,  let  the 
point  M  be  placed  in  any  curve  line.  Therefore  the  point  D  will  be  placed 
in  a  conic  section  passing  through  the  five  points  B,  C,  A,  p,  P,  when  the 
point  M  is  perpetually  placed  in  a  curve  line.  But  from  what  was  de 
monstrated  before,  the  point  D  will  be  also  placed  in  a  conic  section  pass 
ing  through  the  same  five  points  B,  C,  A,  p,  P,  when  the  point  M  is  per 
petually  placed  in  a  right  line.  Wherefore  the  two  conic  sections  will  both 
pass  through  the  same  five  points,  against  Corol.  3,  Lem.  XX.  It  is 
therefore  absurd  to  suppose  that  the  point  M  is  placed  in  a  curve  line. 
QE.D. 

PROPOSITION  XXII.     PROBLEM  XIV. 

To  describe  a  trajectory  that  shall  pass  through  Jive  given  points. 
Let  the  five  given  points  be  A,  B,  C,  P,  D.  c 
From  any  one  of  them,  as   A,  to   any  other     sv 
two  as  B,  C,  which  may  be  called  the  poles, 
draw  the  right  lines  AB,  AC,  and  parallel  to 
those  the  lines  TPS,  PRO,  through  the  fourth 
point   P.     Then  from   the  two  poles  B,  C, 
draw  through  the  fifth  point  D  two  indefinite 
lines  BDT,  CRD,  meeting  with  the  last  drawn  lines  TPS,  PRQ  (the 


138 


THE   MATHEMATICAL    PRINCIPLES 


IBOOK    L 


former  with  the  former,  and  the  latter  with  the  latter)  in  T  and  R.  Then 
drawing  the  right  line  tr  parallel  to  TR,  cutting  off  from  the  right  lines 
PT,  PR,  any  segments  Pt,  Pr,  proportional  to  PT,  PR ;  and  if  through 
their  extremities,  t,  r,  and  the  poles  B,  C,  the  right  lines  lit,  Cr  are  drawn, 
meeting  in  d,  that  point  d  will  be  placed  in  the  trajectory  required.  For 
(by  Lena.  XX)  that  point  d  is  placed  in  a  conic  section  passing  through 
the  four  points  A,  B,  C,  P ;  and  the  lines  R/',  TV  vanishing,  the  point  d 
comes  to  coincide  with  the  point  D.  Wherefore  the  conic  section  passes 
through  the  five  points  A,  B,  C,  P,  D.  Q.E.D. 

The  same  otherwise. 

Of  the  given  points  join  any  three,  as  A,  B, 
C ;  and  about  two  of  them  15,  C,  as  poles, 
making  the  angles  ABC,  ACB  of  a  given 
magnitude  to  revolve,  apply  the  legs  BA, 
CA,  first  to  the  point  D,  then  to  the  point  P, 
and  mark  the  points  M,  N,  in  which  the  other 
legs  BL,  CL  intersect  each  other  in  both  cases.  C' 
Draw  the  indefinite  right  line  MN,  and  let 
those  moveable  angles  revolve  about  their 
poles  B,  C,  in  such  manner  that  the  intersection,  which  is  now  supposed  to 
be  ???,  of  the  legs  BL,  CL;  or  BM7  CM,  may  always  fall  in  that  indefinite 
right  line  MN  ;  and  the  intersection,  which  is  now  supposed  to  be  d,  of  the 
legs  BA  ^A,  or  BD;  CD,  will  describe  the  trajectory  required,  PADc/B. 
For  (by  Lem.  XXI)  the  point  d  will  be  placed  in  a  conic  section  passing- 
through  the  points  B,  C  ;  and  when  the  point  m  comes  to  coincide  with 
the  points  L,  M,  N,  the  point  d  will  (by  construction)  come  to  coin 
cide  with  the  points  A,  D,  P.  Wherefore  a  conic  section  will  be  described 
that  shall  pass  through  the  five  points  A,  B.  C,  P,  D.  Q,.E.F. 

COR.  1.  Hence  a  right  line  may  be  readily  drawn  which  shall  be  a  tan 
gent  to  the  trajectory  in  any  given  point  B.  Let  the  point  d  come  to  co 
incide  with  the  point  B,  arid  the  right  line  Bt/  Avill  become  the  tangent 
required. 

COR.  2.  Hence  also  may  be  found  the  centres,  diameters,  and  latera  recta 
of  the  trajectories,  as  in  Cor.  2,  Lem.  XIX. 

SCHOLIUM. 

The  former  of  these  constructions  will  be-  c 
come  something  more  simple  by  joining  , 
and  in  that  line,  produced,  if  need  be,  aking 
Bp  to  BP  as  PR  is  to  PT ;  and  t  rough  p 
draw  the  indefinite  right  inc  j0e  parallel  to  S 
PT,  and  in  that  line  pe  taking  always  pe 
equal  to  Pi  ,  and  draw  the  right  lines  Be,  Cr 


SEC.  Y.J 


OF    NATURAL    PHILOSOPHY. 


139 


to  meet  in  d.     For  since  Pr  to  Pt,  PR  to  PT,  pB  to  PB,  pe  to  Pt,  are  all  in 

the  same  ratio,  pe  and  Pr  will  be  always  equal.  After  this  manner  the 
points  of  the  trajectory  are  most  readily  found,  unless  you  would  rather 
describe  the  curve  mechanically,  as  in  the  second  construction. 

PROPOSITION  XXIII.     PROBLEM  XV. 

To  describe  a  trajectory  that  shall  pass  through  four  given  points,  and 

touch  a  right  line  given  by  position. 

CASE  1.  Suppose  that  HB  is  the 
given  tangent,  B  the  point  of  contact, 
and  C,  1.,  P,  the  three  other  given 
points.  Jo  n  BC.  and  draw  IS  paral 
lel  to  BH,  and  PQ  parallel  to  BC ; 
complete  the  parallelogram  BSPQ. 
Draw  BD  cutting  SP  in  T,  and  CD 
cutting  PQ,  in  R.  Lastly,  draw  any 
line  tr  parallel  to  TR,  cutting  off 
from  PQ,  PS,  the  segments  Pr,  Pt  proportional  to  PR,  PT  respectively  ; 
and  draw  Cr,  Bt  their  point  of  concourse  d  will  (by  Lem.  XX)  always  fall 
on  the  trajectory  to  be  described. 

The  same  otherwise. 

1  et  tl  e  angle  CBH  of  a  given  magnitude  re 
volve  about  the  pole  B;  as  also  the  rectilinear  ra- 
d:us  1C,  both  ways  produced,  about  the  pole  C. 
Mark  the  points  M,  N,  on  which  the  leg  BC  of 
the  angle  cuts  that  radius  when  BH;  the  other 
leg  thereof,  meets  the  same  radius  in  the  points 
P  and  D.  Then  drawing  the  indefinite  line  MN, 
let  that  radius  CP  or  CD  and  the  leg  BC  of  the 
angle  perpetually  meet  in  this  Ikie;  and  the 
point  of  concourse  of  the  other  leg  BH  with  the 
radius  will  delineate  the  trajectory  required. 

For  if  in  the  constructions  of  the  preceding  Problem  the  point  A  comes 
to  a  coincidence  with  the  point  B,  the  lines  CA  and  CB  will  coincide,  and 
the  line  AB,  in  its  last  situation,  will  become  the  tangent  BH  ;  and  there 
fore  the  constructions  there  set  down  will  become  the  same  with  the  con 
structions  here  described.  Wherefore  the  concourse  of  the  leg  BH  with 
the  radius  will  describe  a  conic  section  passing  through  the  points  C,  D, 
P,  and  touching  the  line  BH  in  the  point  B.  Q.E.F. 

CASE  2.  Suppose  the  four  points  B,  C,  D,  P,  given,  being  situated  with- 
ont  the  tangent  HI.  Join  each  two  by  the  lines  BD,  CP  meeting  in  G, 
and  cutting  the  tangent  in  H  and  I.  Cut  the  tangent  in  A  in  such  mannr: 


140 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK    I 


X 


IT 


that  HA  may  be  to  IA  as  the  rectangle  un 
der  a  mean  proportional  between  CG  and 
GP,  and  a  mean  proportional  between  BH 
and  HD  is  to  a  rectangle  under  a  mean  pro 
portional  between  GD  and  GB,  and  a  mean 
proportional  betweeen  PI  and  1C,  and  A  will 
be  the  point  of  contact.  For  if  HX,  a  par 
allel  to  the  right  line  PI,  cuts  the  trajectory 
in  any  points  X  and  Y,  the  point  A  (by  the 
properties  of  the  conic  sections)  will  come  to  be  so  placed,  that  HA2  will 
become  to  AP  in  a  ratio  that  is  compounded  out  of  the  ratio  of  the  rec 
tangle  XHY  to  the  rectangle  BHD,  or  of  the  rectangle  CGP  to  the  rec 
tangle  DGB;  and  the  ratio  of  the  rectangle  BHD  to  the  rectangle  PIC. 
But  after  the  point  of  contac.t  A  is  found,  the  trajectory  will  be  described  as 
in  the  first  Case.  Q.E.F.  But  the  point  A  may  be  taken  either  between 
or  without  the  points  H  and  I,  upon  which  account  a  twofold  trajectory 
may  be  described. 

PROPOSITION  XXIV.     PROBLEM  XVI. 

To  describe  a  trajectory  that  shall  pass  through  three  given  points,  and 

touch  two  right  lines  given  by  position. 

Suppose  HI,  KL  to  be  the  given  tangents 
and  B,  C,  D,  the  given  points.  Through  any 
two  of  those  points,  as  B,  D,  draw  the  indefi 
nite  right  line  BD  meeting  the  tangents  in 
the  points  H,  K.  Then  likewise  through 
any  other  two  of  these  points,  as  C,  D,  draw 
the  indefinite  right  line  CD  meeting  the  tan 
gents  in  the  points  I,  L.  Cut  the  lines  drawn 
in  R  and  S,  so  that  HR  may  be  to  KR  as 
the  mean  proportional  between  BH  and  HD  is  to  the  mean  proportional 
between  BK  and  KD ;  and  IS  to  LS  as  the  mean  pioportional  between 
CI  and  ID  is  to  the  mean  proportional  between  CL  and  LD.  But  you 
may  cut,  at  pleasure,  either  within  or  between  the  points  K  and  H,  I  and 
L,  or  without  them ;  then  draw  RS  cutting  the  tangents  in  A  and  P,  and 
A  and  P  will  be  the  points  of  contact.  For  if  A  and  P  are  supposed  to 
be  the  points  of  contact,  situated  anywhere  else  in  the  tangents,  and  through 
any  of  the  points  H,  I,  K,  L,  as  I,  situated  in  either  tangent  HI,  a  right 
line  IY  is  drawn  parallel  to  the  other  tangent  KL,  and  meeting  the  curve 
in  X  and  Y,  and  in  that  right  line  there  be  taken  IZ  equal  to  a  mean  pro 
portional  between  IX  and  IY,  the  rectangle  XIY  or  IZ2,  will  (by  the  pro 
perties  of  the  conic  sections)  be  to  LP2  as  the  rectangle  CID  is  to  the  rect 
angle  CLD,  that  is  (by  the  construction),  as  SI  is  to  SL2;  and  therefore 


SEC.  V.]  OF    NATUKAL    PHILOSOPHY.  141 

IZ  is  to  LP  as  SI  to  SL.  Wherefore  the  points  S,  P,  Z.  are  in  one  right 
line.  Moreover,  since  the  tangents  meet  in  G,  the  rectangle  XI Y  or  IZ2 
will  (by  the  properties  of  the  conic  sections)  be  to  IA2  as  GP2  is  to  GA2, 
and  consequently  IZ  will  be  to  I A  as  GP  to  GA.  Wherefore  the  points 
P,  Z,  A,  lie  in  one  right  line,  and  therefore  the  points  S,  P,  and  A  are  in 
one  right  line.  And  the  same  argument  will  prove  that  the  points  R,  P, 
and  A  are  in  one  right  line.  Wherefore  the  points  of  contact  A  and  P  lie 
in  the  right  line  RS.  But  after  these  points  are  found,  the  trajectory  may 
be  described,  as  in  the  first  Case  of  the  preceding  Problem.  Q,.E.F. 

In  this  Proposition,  and  Case  2  of  the  foregoing,  the  constructions  are 
the  same,  whether  the  right  line  XY  cut  the  trajectory  in  X  and  Y,  or 
not ;  neither  do  they  depend  upon  that  section.  But  the  constructions 
being  demonstrated  where  that  right  line  does  cut  the  trajectory,  the  con 
structions  where  it  does  not  are  also  known ;  and  therefore,  for  brevity's 
sake,  I  omit  any  farther  demonstration  of  them. 

LEMMA  XXII. 

To  transform  figures  into  other  figures  of  the  same  kind. 

Suppose  that  any  figure  HGI  is  to  be 
transformed.  Draw,  at  pleasure,  two  par 
allel  lines  AO,  BL,  cutting  any  third  line 
AB,  given  by  position,  in  A  and  B,  and  from 
any  point  G  of  the  figure,  draw  out  any 
right  line  GD,  parallel  to  OA,  till  it  meet 
the  right  line  AB.  Then  from  any  given 
point  0  in  the  line  OA,  draw  to  the  point 
D  the  right  line  OD,  meeting  BL  in  d  ;  and 
from  the  point  of  concourse  raise  the  right 
line  dg  containing  any  given  angle  with  the  right  line  BL,  and  having 
such  ratio  to  Qd  as  DG  has  to  OD ;  and  g  will  be  the  point  in  the  new 
figure  hgi,  corresponding  to  the  point  G.  And  in  like  manner  the  several 
points  of  the  first  figure  will  give  as  many  correspondent  points  of  the  new 
figure.  If  we  therefore  conceive  the  point  G  to  be  carried  along  by  a  con 
tinual  motion  through  all  the  points  of  the  first  figure,  the  point  g  will 
be  likewise  carried  along  by  a  continual  motion  through  all  the  points  of 
the  new  figure,  and  describe  the  same.  For  distinction's  sake,  let  us  call 
DG  the  first  ordinate,  dg  the  new  ordinate,  AD  the  first  abscissa,  ad  the 
new  abscissa ;  O  the  pole.  OD  the  abscinding  radius,  OA  the  first  ordinate 
radius,  and  Oa  (by  which  the  parallelogram  OABa  is  completed)  the  new 
ordinate  radius. 

I  say,  then,  that  if  the  point  G  is  placed  in  a  right  line  given  by  posi 
tion,  the  point  g  will  be  also  placed  in  a  right  line  given  by  position.  If 
the  point  G  is  placed  in  a  conic  section,  the  point  g  will  be  likewise  placed 


J42  THE    MATHEMATICAL    PRINCIPLES  [BOOK    1. 

in  a  conic  section.  And  here  I  understand  the  circle  as  one  of  the  conic 
sections.  But  farther,  if  the  point  G  is  placed  in  a  line  of  the  third  ana 
lytical  order,  the  point  g  will  also  be  placed  in  a  line  of  the  third  order, 
and  so  on  in  curve  lines  of  higher  orders.  The  two  lines  in  which  the 
points  G,  g,  are  placed,  will  be  always  of  the  same  analytical  order.  For 
as  ad  is  to  OA,  so  are  Od  to  OD,  dg  to  DG,  and  AB  to  AD ;  and  there- 

OA  X  AB  OA  X  dg 

fore  AD  is  equal  to , ,  and  DG  equal  to 7— — .    Now  if  the 

ad  ad 

point  G  is  placed  in  a  right  line,  and  therefore,  in  any  equation  by  which 
the  relation  between  the  abscissa  AD  and  the  ordinate  GD  is  expressed, 
those  indetermined  lines  AD  and  DG  rise  no  higher  than  to  one  dimen- 

v         v       xu-  ,.      OA  X  AB  .  OA  X  dg 

sion,  by  writing  this  equation— . m  place  of  AD,  and —   — -. — - 

in  place  of  DG,  a  new  equation  will  be  produced,  in  which  the  new  ab 
scissa  ad  and  new  ordinate  dg  rise  only  to  one  dimension ;  and  which 
therefore  must  denote  a  right  line.  But  if  AD  and  DG  (or  either  of 
them)  had  risen  to  two  dimensions  in  the  first  equation,  ad  and  dg  would 
likewise  have  risen  to  tAvo  dimensions  in  the  second  equation.  And  so  on 
in  three  or  more  dimensions.  The  indetermined  lines,  ad}  dg  in  the 
second  equation,  and  AD,  DG,  in  the  first,  will  always  rise  to  the  same 
number  of  dimensions ;  and  therefore  the  lines  in  which  the  points  G,  g, 
are  placed  are  of  the  same  analytical  order. 

I  say  farther,  that  if  any  right  line  touches  the  curve  line  in  the  first 
figure,  the  same  right  line  transferred  the  same  way  with  the  curve  into 
the  new  figure  will  touch  that  curve  line  in  the  new  figure,  and  vice  versa. 
For  if  any  two  points  of  the  curve  in  the  first  figure  are  supposed  to  ap 
proach  one  the  other  till  they  come  to  coincide,  the  same  points  transferred 
will  approach  one  the  other  till  they  come  to  coincide  in  the  new  figure ; 
and  therefore  the  right  lines  with  which  those  points  are  joined  will  be 
come  together  tangents  of  the  curves  in  both  figures.  I  might  have  given 
demonstrations  of  these  assertions  in  a  more  geometrical  form ;  but  I  study 
to  be  brief. 

Wherefore  if  one  rectilinear  figure  is  to  be  transformed  into  another,  we 
need  only  transfer  the  intersections  of  the  right  lines  of  which  the  first 
figure  consists,  and  through  the  transferred  intersections  to  draw  right  lines 
in  the  new  figure.  But  if  a  curvilinear  figure  is  to  be  transformed,  we 
must  transfer  the  points,  the  tangents,  and  other  right  lines,  by  means  of 
which  the  curve  line  is  denned.  This  Lemma  is  of  use  in  the  solution  of 
the  more  difficult  Problems ;  for  thereby  we  maj'  transform  the  proposed 
figures,  if  they  are  intricate,  into  others  that  are  more  simple.  Thus  any 
right  lines  converging  to  a  point  are  transformed  into  parallels,  by  taking 
for  the  first  ordinate  radius  any  right  line  that  passes  through  the  point 
of  concourse  of  the  converging  lines,  and  that  because  their  point  of  con- 


SEC.  V.]  OF    NATURAL    PHILOSOPHY.  143 

course  is  by  this  means  made  to  go  off  in  infinitum  ;  and  parallel  lines 
are  such  as  tend  to  a  point  infinitely  remote.  And  after  the  problem  is 
solved  in  the  new  figure,  if  by  the  inverse  operations  we  transform  the 
new  into  the  first  figure,  we  shall  have  the  solution  required. 

This  Lemma  is  also  of  use  in  the  solution  of  solid  problems.  For  as 
often  as  two  conic  sections  occur,  by  the  intersection  of  which  a  problem 
may  be  solved,  any  one  of  them  may  be  transformed,  if  it  is  an  hyperbola 
or  a  parabola,  into  an  ellipsis,  and  then  this  ellipsis  may  be  easily  changed 
into  a  circle.  So  also  a  right  line  and  a  conic  section,  in  the  construc 
tion  of  plane  problems,  may  be  transformed  into  a  right  line  and  a  circle 

PROPOSITION  XXV.     PROBLEM   XVII. 

To  describe  a  trajectory  that  shall  pass  through  two  given  points,  and 

touch  three  right  lines  given  by  position. 

Through  the  concourse  of  any  two  of  the  tangents  one  with  the  other, 
and  the  concourse  of  the  third  tangent  with  the  right  line  which  passes 
through  the  two  given  points,  draw  an  indefinite  right  line ;  and,  taking 
this  line  for  the  first  ordinate  radius,  transform  the  figure  by  the  preceding 
Lemma  into  a  new  figure.  In  this  figure  those  two  tangents  will  become 
parallel  to  each  other,  and  the  third  tangent  will  be  parallel  to  the  right 
line  that  passes  through  the  two  given  points.  Suppose  hi,  kl  to  be  those 
two  parallel  tangents,  ik  the  third  tangent,  and  hi  a  right  line  parallel 
thereto,  passing  through  those  points  a,  b, 
through  which  the  conic  section  ought  to  pass 
in  this  new  figure;  and  completing  the  paral- 
lelogra  n  fiikl,  let  the  right  lines  hi,  ik,  kl  be 
BO  cut  in  c,  d,  e,  that  he  may  be  to  the  square 
root  of  the  rectangle  ahb,  ic,  to  id,  and  ke  to 
kd.  as  the  sum  of  the  right  lines  hi  and  kl  is 
to  the  sum  of  the  three  lines,  the  first  whereof ' 
is  the  right  line  ik,  and  the  other  two  are  the 

square  roots  of  the  rectangles  ahb  and  alb  ;  and  c,  d,  e,  will  be  the  points 
of  contact.  For  by  the  properties  of  the  conic  sections,  he2  to  the  rectan 
gle  ahb,  and  ic2  to  id2,  and  ke2  to  kd2,  and  el2  to  the  rectangle  alb,  are  all 
in  the  same  ratio ;  and  therefore  he  to  the  square  root  of  ahb,  ic  to  id,  ke 
to  kdj  and  el  to  the  square  root  of  alb,  are  in  the  subduplicate  of  that 
ratio ;  and  by  composition,  in  the  given  ratio  of  the  sum  of  all  the  ante 
cedents  hi  +  kly  to  the  sum  of  all  the  consequents  ^/ahb  -\-  ik  :  *Jalb, 
Wherefore  from  that  given  ratio  we  have  the  points  of  contact  c,  d,  e,  in 
the  new  figure.  By  the  inverted  operations  of  the  last  Lemma,  let  those 
points  be  transferred  into  the  first  figure,  and  the  trajectory  will  be  there 
described  by  Prob.  XIV.  Q.E.F.  But  according  as  the  points  a,  b,  fall 
between  the  points  //,  /,  or  without  taem,  the  points  c,  d,  e,  must  be  taken 


144  THE    MATHEMATICAL    PRINCIPLES  BOOK  I.J 

Cither  between  the  points,  h,  i,  k,  /,  or  without  them.  If  one  of  the  points 
a,  b,  falls  between  the  points  h,  i,  and  the  other  xvithout  the  points  h,  I, 
the  Problem  is  impossible. 

PROPOSITION  XXVI.     PROBLEM  XVIII. 

To  describe  a  trajectory  that  shall  pass  through  a  given  point,  and  touch 

four  right  lines  given  by  position. 

From  the  common  intersections,  of  any  two 
of  the  tangents  to  the  common  intersection  of 
the  other  two,  draw  an  indefinite  right  line ;  and 
taking  this  line  for  the  first  ordinate  radius;  /  'xs  o 
transform  the  figure  (by  Lem.  XXII)  into  a  new 
figure,  and  the  two  pairs  of  tangents,  each  of 
which  before  concurred  in  the  first  ordinate  ra- 


dius,  will  now  become  parallel.     Let  hi  and  kl,  Al  l\ 

ik  and  hi,  be  those  pairs  of  parallels  completing  the  parallelogram  hikl. 
And  let  p  be  the  point  in  this  new  figure  corresponding  to  the  given  point 
in  the  first  figure.  Through  O  the  centre  of  the  figure  draw  pq.:  and  O? 
being  equal  to  Op,  q  will  be  the  other  point  through  which  the  conic  sec 
tion  must  pass  in  this  new  figure.  Let  this  point  be  transferred,  by  the 
inverse  operation  of  Lem.  XXII  into  the  first  figure,  and  there  we  shall 
have  the  two  points  through  which  the  trajectory  is  to  be  described.  But 
through  those  points  that  trajectory  may  be  described  by  Prop.  XVII. 


LEMMA  XXIII. 

If  two  right  lines,  as  AC,  BD  given  by  position,  and  terminating  in 

given  points  A,  B,  are  in  a  given  ratio  one  to  the  other,  and  the  right 

line  CD,  by  which  the,  indetermined  points  C,  D  are  joined  is  cut  in 

K  in  a  given  ratio  ;  I  say,  that  the  point  K  will  be  placed  in  a  right 

line  given  by  position. 

For  let  the  right  lines  AC,  BD  meet  in 

E,  and  in  BE  take  BG  to  AE  as  BD  is  to 

AC,  and  let  FD  be  always  equal  to  the  given 

line  EG ;  and,  by  construction,  EC  will  be 

to  GD,  that  is,  to  EF,  as  AC  to  BD,  and 

therefore  in  a  given  ratio  ;  and  therefore  the      %•-'''' ,.---''         I  \ 

triangle  EFC  will  be  given  in  kind.  Let  E  K  cT^"^ 
CF  be  cut  in  L  so  as  CL  may  be  to  CF  in  the  ratio  of  CK  to  CD ;  and 
because  that  is  a  given  ratio,  the  triangle  EFL  will  be  given  in  kind,  and 
therefore  the  point  L  will  be  placed  in  the  right  line  EL  given  by  position. 
Join  LK,  and  the  triangles  CLK,  CFD  will  be  similar ;  and  because  FD 
is  a  given  line,  and  LK  is  to  FD  in  a  given  ratio,  LK  will  be  also  given 


SEC.    V.]  OF    NATURAL    PHILOSOPHY.  145 

To  this  let  EH  be  taken  equal,  and  ELKH  will  be  always  a  parallelogram. 
And  therefore  the  point  K  is  always  placed  in  the  side  HK  (given  by  po 
tiition)  of  that  parallelogram.     Q.E.D. 

COR.  Because  the  figure  EFLC  is  given  in  kind,  the  three  right  lines 
EF,  EL,  and  EC,  that  is,  GD,  HK,  and  EC,  will  have  given  ratios  to 
each  other. 

LEMMA  XXIV. 

If  three  right  lines,  two  whereof  are  parallel,  and  given  by  position,  touch 
any  conic  section  ;  I  say,  that  the  semi-diameter  of  the  section  wkiJt 
is  parallel  to  those  two  is  a  mean  proportional  between  the  segments 
of  those  two  that  are  intercepted  between  the  points  of  contact  and  the. 
third  tangent. 

Let  AF,  GB  be  the  two  parallels  touch 
ing  the  conic  section  ADB  in  A  and  B  ; 
EF  the  third  right  line  touching  the  conic 
section  in  I,  and  meeting  the  two  former 
tangents  in  F  and  G,  and  let  CD  be  the 
semi-diameter  of  the  figure  parallel  to 
those  tangents ;  I  say.  that  AF,  CD,  BG 
are  continually  proportional. 

For  if  the  conjugate  diameters  AB,  DM      G  Q 

meet  the  tangent  FG  in  E  and  H,  and  cut  one  the  other  in  C;  and  the 
parallelogram  IKCL  be  completed  ;  from  the  nature  of  the  conic  sections, 
EC  will  be  to  CA  as  CA  to  CL  ;  and  so  by  division,  EC  —  CA  to  CA  -— 
CL,  orEAto  AL;  and  by  composition,  EA  to  EA  +  AL  or  EL,  as  EC  to 
EC  +  CA  or  EB  ;  and  therefore  (because  of  the  similitude  of  the  triangles 
EAF,  ELI,  ECH,  EBG)  AF  is  to  LI  as  CH  to  BG.  Likewise,  from  tli? 
nature  of  the  conic  sections,  LI  (or  CK)  is  to  CD  as  CD  to  CH  ;  and 
therefore  (ex  aquo  pertnrhatfy  AF  is  to  CD  as  CD  to  BG.  Q.E.D. 

COR.  1.  Hence  if  two  tangents  FG,  PQ,  meet  two  parallel  tangents  AF, 
BG  in  F  and  G,  P  and  Q,,  and  cut  one  the  other  in  O;  AF  (ex  cequo  per- 
tnrbot, )  will  be  to  BQ  as  AP  to  BG,  and  by  division,  as  FP  to  GQ,  and 
therefore  as  FO  to  OG. 

COR.  2.  Whence  also  the  two  right  lines  PG,  FQ,  drawn  through  the 
points  P  and  G,  F  and  Q,  will  meet  in  the  right  line  ACB  passing  through 
the  centre  of  the  figure  and  the  points  of  contact  A,  B. 

LEMMA  XXV. 

If four  sides  of  a  parallelogram  indefinitely  produced  touch  any  conic 
section,  and  are  cut  by  a  fifth  tangent ;  I  say,  that,  taking  those  seg 
ments  of  any  two  conterminous  sides  that  terminate  in  opposite  angles 

10 


146 


THE    MATHEMATICAL    PRINCIPLES 


[BooK  1. 


of  the  parallelogram,  either  segment  is  to  the  side  from  which  it  is 

cut  off  as  that  part  of  the  other  conterminous  side  which  is  intercepted 

between  the  point  of  contact  and  the  third  side  is  to  Uie  other  segment, 

Let  the  four  sides  ML,  IK,  KL,  MI, 
of  the  parallelogram  ML  JK  touch  the 
conic  section  in  A,  B,  C,  I)  ;  and  let  the 
fifth  tangent  FQ  cut  those  sides  in  F, 
Q,  H,  and  E  :  and  taking  the  segments 
ME,  KQ  of  the  sides  Ml,  KJ,  or  the 
segments  KH,  MF  of  the  sides  KL, 
ML,  1  s/.y,  that  ME  is  to  MI  as  BK  to 
KQ;  and  KH  to  KL  as  AM  to  MF. 
For,  by  Cor.  1  of  the  preceding  Lemma,  ME  is  to  El  as  (AM  or)  BK  to 
BQ  ;  and,  by  composition,  ME  is  to  MI  as  BK  to  KQ.  Q.E.D.  Also 
KH  is  to  HL  as  (BK  or)  AM  to  AF  ;  and  by  division,  KH  to  KL  as  AM 
to  MF.  Q.E.D. 

COR.  1.  Hence  if  a  parallelogram  IKLM  described  about  a  given  conic 
section  is  given,  the  rectangle  KQ  X  ME,  as  also  the  rectangle  KH  X  ME 
equal  thereto,  will  be  given.  For,  by  reason  of  the  similar  triangles  KQH 
MFE,  those  rectangles  are  equal. 

COR.  2.  And  if  a  sixth  tangent  eq  is  drawn  meeting  the  tangents  Kl. 
MI  in  q  and  e,  the  rectangle  KQ  X  ME  will  be  equal  to  the  rectangle 
K</  X  Me,  and  KQ  will  be  to  Me  as  Kq  to  ME,  and  by  division  ns 
Q?  to  Ee. 

COR.  3.  Hence,  also,  if  E<?,  eQ,  are  joined  and  bisected,  and  a  right  line 
is  drawn  through  the  points  of  bisection,  this  right  line  will  pass  through 
the  centre  of  the  conic  section.  For  since  Q</  is  to  Ee  as  KQ  to  Me,  the 
same  right  line  will  pass  through  the  middle  of  all  the  lines  Eq,  eQ,  MK 
(by  Lem.  XXIII),  and  the  middle  point  of  the  right  line  MK  is  the 
centre  of  the  section. 

PROPOSITION  XXVII.    PROBLEM  XIX. 

To  describe  a  trajectory  that  may  touch  jive  right  lines  given  by  position. 

Supposing    ABG;   BCF, 
GCD,  FDE,  EA  to  be  the 

tangents  given  by  position. 
Bisect  in  M  and  N,  AF,  BE, 
the  diagonals  of  the  quadri 
lateral 
tained 


figure 
under 


ABFE  con- 
any  four  of 
them  ;  and  (by  Cor.  3,  Lem. 
XXV)  the  right  line  MN 
draAvn  through  the  points  (,f 


SEC.  V.]  OF    NATURAL    PHILOSOPHY.  147 

bisection  will  pass  through  the  centre  of  the  trajectory.  Again,  bisect  in 
P  and  Q,  the  diagonals  (if  I  may  so  call  them)  Bl),  GF  of  the  quadrila 
teral  figure  EC  OF  contained  under  any  other  four  tangents,  and  the  right 
line  PQ,  drawn  through  the  points  of  bisection  will  pass  through  the  cen 
tre  of  the  trajectory ;  and  therefore  the  centre  will  be  given  in  the  con 
course  of  the  bisecting  lines.  Suppose  it  to  be  O.  Parallel  to  any  tan 
gent  BG  draw  KL  at  such  distance  that  the  centre  O  may  be  placed  in  the 
middle  between  the  parallels;  this  KL  will  touch  the  trajectory  to  be  de 
scribed.  Let  this  cut  any  other  two  tangents  GCD,  FJ)E,  in  L  and  K. 
Through  the  points  G  and  K,  F  and  L,  where  the  tangents  not  parallel, 
CL,  FK  meet  the  parallel  tangents  CF,  KL,  draw  GK,  FL  meeting  in 
K ;  and  the  right  line  OR  drawn  and  produced,  will  cut  the  parallel  tan 
gents  GF,  KL,  in  the  points  of  contact.  This  appears  from  Gor.  2,  Lem. 
XXIV.  And  by  the  same  method  the  other  points  of  contact  may  be 
found,  and  then  the  trajectory  may  be  described  by  Prob.  XIV.  Q.E.F. 

SCPIOLTUM. 

Under  the  preceding  Propositions  are  comprehended  those  Problems 
wherein  either  the  centres  or  asymptotes  of  the  trajectories  are  given.  For 
when  points  and  tangents  and  the  centre  are  given,  as  many  other  points 
and  as  many  other  tangents  are  given  at  an  equal  distance  on  the  other 
side  of  the  centre.  And  an  asymptote  is  to  be  considered  as  a  tangent,  ami 
its  infinitely  remote  extremity  (if  we  may  say  so)  is  a  point  of  contact. 
Conceive  the  point  of  contact  of  any  tangent  removed  in  infinitum,  and 
the  tangent  will  degenerate  into  an  asymptote,  and  the  constructions  of 
the  preceding  Problems  will  be  changed  into  the  constructions  of  those 
Problems  wherein  the  asymptote  is  given. 

After  the  trajectory  is  described,  we  may 
find  its  axes  and  foci  in  this  manmr.  In  the 
construction  and  figure  of  Lem.  XXI,  let  those  , 

legs  BP,  CP,  of  the  moveable  angles  PEN,  ^ 
PCN,  by  the  concourse  of  which  the  trajec-  \ 
tory  was  described,  be  made  parallel  one  to 
the  other :  and  retaining  that  position,  let 
them  revolve  about  their  poles  I',  C,  in  that 
figure.  In  the  mean  while  let  the  other  legs 
GN,  BN,  of  those  angles,  by  their  concourse 
K  or  k,  describe  the  circle  BKGC.  Let  O  be  the  centre  of  this  circle; 
and  from  this  centre  upon  the  ruler  MN,  wherein  those  legs  CN,  BN  did 
concur  while  the  trajectory  was  described,  let  fall  the  perpendicular  OH 
meeting  the  circle  in  K  and  L.  And  when  those  other  legs  CK,  BK  meet 
in  the  point  K  that  is  nearest  to  the  ruler,  the  first  legs  CP,  BP  will  be 
parallel  to  the  greater  axis,  and  perpendicular  on  the  lesser ;  and  the  con- 


148 


THE    MATHEMATICAL    PRINCIPLES 


[Book   I 


trary  will  hajpen  if  those  legs  meet  in  the  remotest  point  L.  'Whence  ii 
the  centre  of  the  trajectory  is  given,  the  axes  will  be  given ;  and  those  be- 
ing  given,  the  foci  will  be  readily  found. 

But  the  squares  of  the  axes  are  one  to  the 
other  as  KH  to  LH,  and  thence  it  is  easy  to 
describe  a  trajectory  given  in  kind  through 
f  mr  given  points.  For  if  two  of  the  given 
points  are  made  the  poles  C,  13,  the  third  will 
give  the  moveable  angles  PCK,  PBK ;  but 
those  being  given,  the  circle  BGKC  may  be 
described.  Then,  because  the  trajectory  is 
given  in  kind,  the  ratio  of  OH  to  OK,  and 
and  therefore  OH  itself,  will  be  given.  About  ' 
the  centre  O,  with  the  interval  OH,  describe  another  circle,  and  the  right 
line  that  touches  this  circle,  and  passes  through  the  concourse  of  the  legs 
CK,  BK,  when  the  first  legs  CP;  BP  meet  in  the  fourth  given  point,  will 
be  the  ruler  MN,  by  means  of  which  the  trajectory  may  be  described 
Whence  also  on  the  other  hand  a  trapezium  given  in  kind  (excepting  a 
few  cases  that  are  impossible)  may  be  inscribed  in  a  given  conic  section. 

There  are  also  other  Lemmas,  by  the  help  of  which  trajectories  given  in 
kind  may  be  described  through  given  points,  and  touching  given  lines. 
Of  such  a  sort  is  this,  that  if  a  right  line  is  drawn  through  any  point 
given  by  position,  that  may  cut  a  given  conic  section  in  two  points,  and 
the  distance  of  the  intersections  is  bisected,  the  point  of  bisection  will 
to'ich  ano  her  conic  section  of  the  same  kind  with  the  former,  arid  havin^ 
its  axes  parallel  to  the  axes  of  the  former.  But  I  hasten  to  things  of 
greater  use. 


LEMMA  XXVI. 

To  place  1ht'  lit  rev  angles  of  a  triangle,  given  both  in  kind  and  magni 
tude,  in,  respect  of  as  many  rigid  lines  given  by  position,  -provided  th°\] 
are  not  all  parallel  among  themselves,  in  such  manner  tfia'  tjic  spiral 
angles  may  touch  the  several  lines. 

Three  indefinite  right  lines  AB,  AC,  BC,  are 
given  by  position,  and  it  is  required  so  to  place 
the  triangle  DEF  that  its  angle  1)  may  touch 
the  line  AB,  its  angle  E  the  line  AC,  and 
its  angle  F  the  line  BC.  Upon  DE,  DF,  and 
EF,  describe  three  segments  of  circles  DRE, 
DGF.  EMF,  capable  of  angles  equal  to  the 
Rubles  BAG,  ABC,  ACB  respectively.  But  those  segments  are  to  be  de 
scribed  t«  wards  such  sides  of  the  lines  DE,  DF;  EF;  that  the  letters 


3  EC.   V.I 


OF    NATURAL    PHILOSOPHY. 


1411 


DRED  may  turn  round  about  in  the  same  order  with  the  letters  I1ACB  : 
the  letters  DGFD  in  the  same  order  with  the  letters  ABCA ;  and  the 
letters  EMFE  in  the  same  order  with  the  letters  ACBA ;  then,  completing 
th 'se  segmerts  into  entire  circles  let  the  two  former  circles  cut  one  the 
other  in  G,  and  suppose  P  and  Q  to  be  their  centres.  Then  joining  GP, 
PQ,  take  Ga  to  AB  as  GP  is  to  PQ ;  and  about  the  centre  G,  with  the 
interval  Ga,  describe  a  circle  that  may  cut  the  first  circle  DGE  in  a. 
Join  aD  cutting  the  second  circle  DFG  in  b,  as  well  as  aE  cutting  the 
third  circle  EMF  in  c.  Complete  the  figure  ABCdef  similar  and  equal 
to  the  figure  a&cDEF :  I  say,  the  thing  is  done. 

For  drawing  Fc  meeting  «D  in  n, 
and  joining  aG;  bG,  QG,  QD.  PD,  by 
construction  the  angle  EaD  is  equal  to 
the  angle  CAB,  and  the  angle  acF  equal 
to  the  angle  ACB;  and  therefore  the 
triangle  aiic  equiangular  to  the  triangle 
ABC.  Wherefore  the  angle  anc  or  FnD 
is  equal  to  the  angle  ABC,  and  conse- 
<  uently  to  the  angle  F/>D ;  and  there 
fore  the  point  n  falls  on  the  point  b, 
Moreover  the  angle  GPQ,  which  is  half 
the  angle  GPD  at  the  centre,  is  equal 
to  the  angle  GaD  at  the  circumference  \ 
and  the  angle  GQP,  which  is  half  the  angle  GQD  at  the  centre,  is  equal 
to  the  complement  to  two  right  angles  of  the  angle  GbD  at  the  circum 
ference,  and  therefore  equal  to  the  angle  Gba.  Upon  which  account  the 
triangles  GPQ,  Gab,  are  similar,  and  Ga  is  to  ab  as  GP  to  PQ. ;  that  is 
(by  construction),  as  Ga  to  AB.  Wherefore  ab  and  AB  are  equal;  and 
consequently  the  triangles  abc,  ABC,  which  we  have  now  proved  to  be 
similar,  are  also  equal.  And  therefore  since  the  angles  I),  E,  F,  of  the 
triangle  DEF  do  respectively  touch  the  sides  ab,  ar,  be  of  the  triangle 
afjc/  the  figure  AECdef  may  be  completed  similar  and  equal  to  the  figure 
afrcDEFj  and  by  completing  it  the  Problem  will  be  solved.  Q.E.F. 

COR.  Hence  a  right  line  may  be  drawn  whose  parts  given  in  length  may 
be  intercepted  between  three  right  lines  given  by  position.  Suppose  the 
triangle  DEF,  by  the  access  of  its  point  D  to  the  side  EF,  arid  by  having 
the  sides  DE,  DF  placed  i>t  directum  to  be  changed  into  a  right  line 
whose  given  part  DE  is  to  be  interposed  between  the  right  lines  AB;  AC 
given  by  position;  and  its  given  part  DF  is  to  be  interposed  between  the 
right  lines  AB;  BC,  given  by  position;  then,  by  applying  the  preceding 
construction  to  this  case,  the  Problem  will  be  solved. 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK  1. 


PROPOSITION  XXVIII.     PROBLEM  XX. 

To  describe  a  trajectory  giren  both  in  kind  and  magnitude,  given  parts 
of  which  shall  be  interposed  between  three  right  lines  given  by  position. 
Suppose  a  trajectory  is  to  be  described  that 

may  be  similar  and  equal  to  the  curve  line  DEF, 
-and  may  be  cut  by  three  right  lines  AB,  AC, 

BC,  given  by  position,  into  parts  DE  and  EF, 

similar  and   equal  to  the  given  parts  of  this 

curve  line. 

Draw  the  right  lines  DE,  EF,  DF:    and 

place  the  angles  D,  E,  F,  of  this  triangle  DEF,  so 

as  to  touch  those  right  lines  given  by  position  (by 

Lem.  XXVI).     Then  about  the  triangle   describe 

the  trajectory,  similar  and  equal  to  the  curve  DEF. 

Q.E.F. 

LEMMA  XXVII. 

To  describe  a  trapezium  given  in  kind,  the  angles  whereof  may  be  », 
placed,  in  respect  of  four  right  lines  given  by  position,  that  are  neither 
all  paralhl  among  themselves,  nor  converge  to  one  common  point,  ////// 
the  several  angles  may  touch  the  several  lines. 
Let  the  four  right  lines  ABC,  AD,  BD,  CE,  be 
given  by  position  ;  the  first  cutting  the  second  in  A, 
the  third  in  B,  and  the  fourth  in  C  •  and  suppose  a 
trapezium  fghi  is  to  be  described  that  may  be  similar 
to  the  trapezium  FCHI,  and  whose  angle  /,  equal  to 
the  given  angle  F,  may  touch  the  right  line  ABC ;  and 
(lie  other  angles  g,  h,  i,  equal  to  the  other  given  angles, 
G,  H,  I,  may  touch  the  other  lines  AD,  BD,  CE,  re 
spectively.  Join  FH,  and  upon  FG.  FH,  FI  describe  J% 
as  many  segments  of  circles  FSG,  FTH,  FVI,  the  first 
of  which  FSG  may  be  capable  of  an  angle  equal  to 
the  angle  BAD  ;  the  second  FTH  capable  of  an  angle 
equal  to  the  angle  CBD  ;  and  the  third  FVI  of  an  angle  equal  to  the  angle 
ACE.  Bnrf>,  the  segments  are  to  be  described  towards  those  sides  of  the 
lines  FG,  FH,  FI,  that  the  circular  order  of  the  letters  FSGF  may  be 
the  same  as  of  the  letters  BADB,  and  that  the  letters  FTHF  may  turn 
.ibout  in  the  same  order  as  the  letters  CBDC  and  the  letters  FVIF  in  the 
game  order  as  the  letters  ACE  A.  Complete  the  segments  into  entire  cir 
cles,  and  let  P  be  the  centre  of  the  first  circle  FSG,  Q,  the  centre  of  the 
second  FTH.  Join  and  produce  both  ways  the  line  PQ,,  and  in  it  take 
QR  in  the  same  ratio  to  PQ  as  BC  has  to  AB.  But  QR  is  to  be  taken 
towards  that  side  of  the  point  Q  that  the  order  of  the  letters  P,  Q,,  R 


SEC.  V.J 


OF    NATURAL    PHILOSOPHY. 


15] 


may  be  the  same  as  of  the  letters  A,  B,  C ; 
and  about  the  centre  R  with  the  interval 
RF  describe  a  fourth  circle  FNc  cutting 
(lie  third  circle  FVI  in  c.  Join  Fc1  cut 
ting  the  first  circle  in  a,  and  the  second  in 
/ .  Draw  aG,  &H,  cl,  and  let  the  figure 
ABC/'4f/ii  be  made  similar  to  the  figure 
w^cFGHI;  and  the  trapezium  fghi  will 
be  that  which  was  required  to  be  de 
scribed. 

For  let  the  two  first  circles  FSG,  FTH 
cut  one  the  other  in  K  ;  join  PK,  Q,K, 
RK,  "K,  6K,  cK,  and  produce  QP  to  L. 
The  angles  FaK,  F6K,  FcK  at  the  circumferences  are  the  halves  of  the 
angles  FPK,  FQK,  FRK,  at  the  centres,  and  therefore  equal  to  LPK, 
LQ.K,  LRK,  the  halves  of  those  angles.  Wherefore  the  figure  PQRK  is 
iquiangular  and  similar  to  the  figure  «6cK,  and  consequently  ab  is  to  be 
res  PQ,  to  Q,R,  that  is,  as  AB  to  BC.  But  by  construction,  the  angles 
''Air, /B//,/C?',  are  equal  to  the  angles  F«G,  F&H,  Fcl.  And  therefore 
the  figure  ABCfghi  may  be  completed  similar  to  the  figure  abcFGHl. 
vVliich  done  a  trapezium  fghi  will  be  constructed  similar  to  the  trapezium 
FGHI,  and  which  by  its  angles/,  g,  h,  i  will  touch  the  right  lines  ABC, 
AD,  BD,  CE.  Q.E.F. 

COR.  Hence  a  right  line  may  be  drawn  whose  parts  intercepted  in  a 
given  order,  between  four  right  lines  given  by  position,  shall  have  a  given 
proportion  among  themselves.  Let  the  angles  FGH,  GHI,  be  so  far  in 
creased  that  the  right  lines  FG,  GH,  HI,  may  lie  in  directum  ;  and  by 
constructing  the  Problem  in  this  case,  a  right  line  fghi  will  be  drawn, 
whose  parts  fg,  gh,  hi,  intercepted  between  the  four  right  lines  given  by 
position,  AB  and  AD,  AD  and  BD,  BD  and  CE,  will  be  one  to  another 
as  the  lines  FG,  GH,  HI,  and  will  observe  the  same  order  among  them 
selves.  But  the  same  thing  may  be  more  readily  done  in  this  manner. 

Produce  AB  to  K  and  BD  to  L, 
so  as  BK  may  be  to  AB  as  HI  to 
GH  ;  and  DL  to  BD  as  GI  to  FG; 
and  join  KL  meeting  the  right  line 
CE  in  i.  Produce  iL  to  M,  so  as 
LM  may  be  to  iL  as  GH  to  HI ; 
then  draw  MQ,  parallel  to  LB,  and 
meeting  the  right  line  AD  in  g,  and 
join  gi  cutting  AB,  BD  in  f,  h ;  I  M* 
say,  the  thing  is  done. 

For  let  MO-  cut  the  right  line  AB  in  Q,  and  AD  the  right  line  KL  iu 


II 


^52  THE   MATHEMATICAL    PRINCIPLES  [BOOK    I. 

S,  arid  draw  AP  parallel  to  BD,  and  meeting  iL  in  P,  and  §-M  to  Lh  (g\ 
to  hi,  Mi  to  Li,  GI  to  HI,  AK  to  BK)  and  AP  to  BL,  will  be  in  the  same 
ratio.  Cut  DL  in  11,  so  as  DL  to  RL  may  be  in  that  same  ratio;  and  be 
cause  ffS  to  g~M,  AS  to  AP.  and  DS  to  DL  are  proportional;  therefore 
(ex  ceqit.o)  as  gS  to  LA,  so  will  AS  be  to  BL,  and  DS  to  RL ;  and  mixtly. 
BL  —  RL  to  Lh  —  BL,  as  AS  —  DS  to  gS  —  AS.  That  is,  BR  is  to 
Eh  as  AD  is  to  Ag,  and  therefore  as  BD  to  gQ.  And  alternately  BR  is 
to  BD  as  13/i  to  g-Q,,  or  as  fh  to  fg.  But  by  construction  the  line  BL 
was  cut  in  D  and  R  in  the  same  ratio  as  the  line  FI  in  G  and  H  ;  and 
therefore  BR  is  to  BD  as  FH  to  FG.  Wherefore  fh  is  to  fg  as  FH  to 
FG.  Since,  therefore,  gi  to  hi  likewise  is  as  Mi  to  Li,  that  is,  as  GI  to 
HI,  it  is  manifest  that  the  lines  FI,  fi,  are  similarly  cut  in  G  and  H,  g 
and  //..  Q.E.F. 

In  the  construction  of  this  Corollary,  after  the  line  LK  is  drawn  cutting 
CE  in  i,  we  may  produce  iE  to  V,  so  as  EV  may  be  to  Ei  as  FH  to  HI, 
arid  then  draw  V/~  parallel  to  BD.  It  will  come  to  the  same,  if  about  the 
centre  i  with  an  interval  IH,  we  describe  a  circle  cutting  BD  in  X,  and 
produce  iX  to  Y  so  as  iY  may  be  equal  to  IF,  and  then  draw  Yf  parallel 

to  BO. 

Sir  Christopher  Wren  and  Dr.  Wallis  have  long  ago  given  other  solu 
tions  of  this  Problem. 

PROPOSITION  XXIX.     PROBLEM  XXI. 

To  describe  a  trajectory  given  in  kind,  that  may  be  cut  by  four  right 
lines  given  by  position,  into  parts  given  in  order,  kind,  and  proportion. 
Suppose  a  trajectory  is  to  be  described  that  may  be 
similar  to  the  curve  line  FGHI,  and  whose  parts, 
similar  and  proportional  to  the  parts  FG,  GH,  HI  of 
the  other,  may  be  intercepted  between  the  right  lines 
AB  and  AD,  AD,  and  BD,  BD  and  CE  given  by  po 
sition,  viz.,  the  first  between  the  first  pair  of  those  lines, 
the  second  between  the  second,  and  the  third  between 
the  third.  Draw  the  right  lines  FG,  GH,  HI,  FI; 
and  (by  Lem.  XXVII)  describe  a  trapezium  fghi  that 
may  be  similar  to  the  trapezium  FGHI,  and  whose  an 
gles/,  g,  h,  i,  may  touch  the  right  lines  given  by  posi 
tion  AB,  AD,  BD,  CE,  severally  according  to  their  order.  And  then  about 
bins  trapezium  describe  a  trajectory,  that  trajectory  will  be  similar  to  the 
curve  line  FGHI. 

SCHOLIUM. 

This  problem  may  be  likewise  constructed  in  the  following  manner. 
Joining  FG,  GH,  HI,  FI,  produce  GF  to  Y,  and  join  FH,  IG,  and  make 


SEC.  VI 


OF    NATURAL    PHILOSOPHY. 


153 


El 


the  angles  CAK.  DAL  equal  to 
the  angles  PGH,  VFH.  Let 

AK,  AL  meet  the  right  line 
BD  in  K  and  L,  and  thence 
draw  KM,  LN,  of  which  let 
KM  make  the  angle  A  KM  equal 
to  the  angle  CHI,  and  be  itself 
to  AK  as  HI  is  to  GH ;  and  let 

LN  make  the  angle  ALN  equal  to  the  angle  FHI,  and  be  itself 
to  AL  as  HI  to  FH.  But  AK,  KM.  AL,  LN  are  to  be  drawn 
towards  those  sides  of  the  lines  AD,  AK,  AL,  that  the  letters 
OA.KMC,  ALKA,  DALND  may  be  carried  round  in  the  same 
order  as  the  letters  FGHIF ;  and  draw  MN  meeting  the  right  v 
line  CE  in  L  Make  the  angle  iEP  equal  to  the  angle  IGF, 
and  let  PE  be  to  Ei  as  FG  to  GI ;  and  through  P  draw  PQ/'  that  may 
with  the  right  line  ADE  contain  an  angle  PQE  equal  to  the  angle  FIG, 
and  may  meet  the  right  line  AB  in  /,  and  join  fi.  But  PE  and  PQ  arc- 
to  be  drawn  towards  those  sides  of  the  lines  CE,  PE,  that  the  circular 
order  of  the  letters  PEtP  and  PEQP  may  be  the  same  as  of  the  letters 
FGHIF  ;  and  if  upon  the  line/i,  in  the  same  order  of  letters,  and  similar 
to  the  trapezium  FGHI,  a  trapezium /^//.i  is  constructed,  and  a  trajectory 
given  in  kind  is  circumscribed  about  it,  the  Problem  will  be  solved. 

So  far  concerning  the  finding  of  the  orbits.     It  remains  that  we  deter 
mine  the  motions  of  bodies  in  the  orbits  so  found. 


SECTION  VI. 

How  the  motions  are  to  be  found  in  given,  orbits. 

PROPOSITION  XXX.     PROBLEM  XXII. 

To  find  at  any  assigned  time  the  place  of  a  body  moving  in,  a  given 

parabolic  trajectory. 

Let  S  be  the  focus,  and  A  the  principal  vertex  of 
the  parabola;  and  suppose  4AS  X  M  equal  to  the 
parabolic  area  to  be  cut  off  APS,  which  either  was 
described  by  the  radius  SP,  since  the  body's  departure 
from  the  vertex,  or  is  to  be  described  thereby  before 
its  arrival  there.  Now  the  quantity  of  that  area  to 
be  cut  off  is  known  from  the  time  which  is  propor 
tional  to  it.  Bisect  AS  in  G,  and  erect  the  perpendicular  GH  equal  to 
3M,  and  a  circle  described  about  th  j  centre  H,  with  the  interval  HS,  will 
cut  the  parabola  in  the  place  P  required.  For  letting  fall  PO  perpendic 
ular  on  the  axis,  and  drawing  PH,  there  will  be  AG2  -f-  GH2  (=.=  HP2  -_ 

AO^TAGJ*  +  PO  —  GH|2)  =  AO2  +  PO2  —  2CA  >  —  ?G!I  f  PO   ' 


A  G  S 


154  THE    MATHEMATICAL    PRINCIPLES  [BOOK    I 

AG*  +  GH2.     Whence  2GH  X  PO  («  AO2  +  PO2  —  2GAO)  =  AOJ 

PO2 
-f  |  PO2.     For  AO2  write  AO  X';   then  dividing  all  the  terms  by 


2PO;  and  multiplying  them  by  2AS,  we  shall  have  ^GH  X  AS  (=  IAO 


the  area  APO  —  SPO)|  =  to  the  area  APS.     But  GH  was  3M,  and 

therefore  ^GH  X  AS  is  4AS  X  M.  Wherefore  the  area  cut  off  APS  is 
equal  to  the  area  that  was  to  be  cut  off  4AS  X  M.  Q.E.D. 

Con.  1.  Hence  GH  is  to  AS  as  the  time  in  which  the  body  described 
the  arc  AP  to  the  time  in  which  the  body  described  the  arc  between  the 
vertex  A  and  the  perpendicular  erected  from  the  focus  S  upon  the  axis. 

COR.  2.  And  supposing  a  circle  ASP  perpetually  to  pass  through  the 
moving  body  P,  the  xelocity  of  the  point  H  is  to  the  velocity  which  the 
body  had  in  the  vertex  A  as  3  to  8;  and  therefore  in  the  same  ratio  is 
the  line  GH  to  the  right  line  which  the  body,  in  the  time  of  its  moving 
from  A  to  P,  would  describe  with  that  velocity  which  it  had  in  the  ver 
tex  A. 

COR.  3.  Hence,  also,  on  the  other  hand,  the  time  may  be  found  in  which 
the  body  has  described  any  assigned  arc  AP.  Join  AP,  and  on  its  middle 
point  erect  a  perpendicular  meeting  the  right  line  GH  in  H, 

LEMMA  XXVIII. 

There  is  no  oval  figure  whose  area,  cut  off  by  right  lines  at  pleasure,  can, 
be  universally  found  by  means  of  equations  of  any  number  of  finite 
terms  and  dimensions. 

Suppose  that  within  the  oval  any  point  is  given,  about  which  as  a  pole 
a  right  line  is  perpetually  revolving  with  an  uniform  motion,  while  in 
that  right  line  a  mov cable  point  going  out  from  the  pole  moves  always 
forward  with  a  velocity  proportional  to  the  square  of  that  right  line  with 
in  the  oval.  By  this  motion  that  point  will  describe  a  spiral  with  infinite 
circumgyrations.  Now  if  a  portion  of  the  area  of  the  oval  cut  off  by  that 
right  line  could  be  found  by  a  finite  equation,  the  distance  of  the  point 
from  the  pole,  which  is  proportional  to  this  area,  might  be  found  by  the 
same  equation,  and  therefore  all  the  points  of  the  spiral  might  be  found 
by  a  finite  equation  also ;  and  therefore  the  intersection  of  a  right  line 
given  in  position  with  the  spiral  might  also  be  found  by  a  finite  equation. 
But  every  right  line  infinitely  produced  cuts  a  spiral  in  an  infinite  num 
ber  of  points  ;  and  the  equation  by  which  any  one  intersection  of  two  lines 
is  found  at  the  same  time  exhibits  all  their  intersections  by  as  many  roots, 
and  therefore  rises  to  as  many  dimensions  as  there  are  intersections.  Be 
cause  two  circles  mutually  cut  one  another  in  two  points,  one  of  those  in- 


8FC.    Vl.J  OF    NATURAL    PHILOSOPHY.  155 

terscctions  is  not  to  be  found  but  by  an  equation  of  two  dimensions,  fo 
which  the  other  intersection  may  be  also  found.  Because  there  may  b(- 
four  intersections  of  two  conic  sections,  any  one  of  them  is  not  to  be  found 
universally,  but  by  an  equation  of  four  dimensions,  by  which  they  may  bi> 
all  found  together.  For  if  those  intersections  are  severally  sought,  be 
cause  the  law  and  condition  of  all  is  the  same,  the  calculus  will  be  the 
same  in  every  case,  and  therefore  the  conclusion  always  the  same,  which 
must  therefore  comprehend  all  those  intersections  at  once  within  itself,  and 
exhibit  them  all  indifferently.  Hence  it  is  that  the  intersections  of  the 
conic  se"f  ions  with  the  curves  of  the  third  order,  because  they  may  amount 
to  six,  (\,me  out  together  by  equations  of  six  dimensions ;  and  the  inter 
sections  of  two  curves  of  the  third  order,  because  they  may  amount  to  nine, 
come  out  together  by  equations  of  nine  dimensions.  If  this  did  not  ne 
cessarily  happen,  we  might  reduce  all  solid  to  plane  Problems,  and  those 
higher  than  solid  to  solid  Problems.  But  here  i  speak  of  curves  irreduci 
ble  in  power.  For  if  the  equation  by  which  the  curve  is  defined  may  bo 
reduced  to  a  lower  power,  the  curve  will  not  be  one  single  curve,  but  com 
posed  of  two,  or  more,  whose  intersections  may  be  severally  found  by  different 
calculusses.  After  the  same  manner  the  two  intersections  of  right  lines 
with  the  conic  sections  come  out  always  by  equations  of  two  dimensions ;  the 
three  intersections  of  right  lines  with  the  irreducible  curves  of  the  third 
urder  by  equations  of  three  dimensions ;  the  four  intersections  of  right 
lines  with  the  irreducible  curves  of  the  fourth  order,  by  equations  of  four 
dimensions  ;  and  so  on  in  iitfinitum.  Wherefore  the  innumerable  inter 
sections  of  a  right  line  with  a  spiral,  since  this  is  but  one  simple  curve 
and  not  reducible  to  more  curves,  require  equations  infinite  in  r-  .imber  of 
dimensions  and  roots,  by  which  they  may  be  all  exhibited  together.  For 
the  law  and  calculus  of  all  is  the  same.  For  if  a  perpendicular  is  let  fall 
from  the  pole  upon  that  intersecting  right  line,  and  that  perpendicular 
together  with  the  intersecting  line  revolves  about  the  pole,  the  intersec 
tions  of  the  spiral  will  mutually  pass  the  one  into  the  other ;  and  that 
which  was  first  or  nearest,  after  one  revolution,  will  be  the  second ;  after 
two,  the  third ;  and  so  on :  nor  will  the  equation  in  the  mean  time  be 
changed  but  as  the  magnitudes  of  those  quantities  are  changed,  by  which 
the  position  of  the  intersecting  line  is  determined.  Wherefore  since  those 
quantities  after  every  revolution  return  to  their  first  magnitudes,  the  equa 
tion  will  return  to  its  first  form ;  and  consequently  one  and  the  same 
equation  will  exhibit  all  the  intersections,  and  will  therefore  have  an  infi 
nite  number  of  roots,  by  which  they  may  be  all  exhibited.  And  therefore 
the  intersection  of  a  right  line  with  a  spiral  cannot  be  universally  found  by 
any  finite  equation ;  and  of  consequence  there  is  no  oval  figure  whose  area, 
cut  off  by  right  lines  at  pleasure,  can  be  universally  exhibited  by  an^ 
such  equation. 


1 56 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK   1 


By  the  same  argument,  if  the  interval  of  the  pole  and  point  by  which 
the  spiral  is  described  is  taken  proportional  to  that  part  of  the  perimeter 
of  the  oval  which  is  cut  off,  it  may  be  proved  that  the  length  of  the  peri 
meter  cannot  be  universally  exhibited  by  any  finite  equation.  But  here  I 
speak  of  ovals  that  are  not  touched  by  conjugate  figures  running  out  in 
infinitvm. 

COR.  Hence  the  area  of  an  ellipsis,  described  by  a  radius  drawn  from 
the  focus  to  the  moving  body,  is  not  to  be  found  from  the  time  given  by  a 
finite  equation  ;  and  therefore  cannot  be  determined  by  the  description  ol 
curves  geometrically  rational.  Those  curves  I  call  geometrically  rational, 
all  the  points  whereof  may  be  determined  by  lengths  that  are  definable 
by  equations ;  that  is,  by  the  complicated  ratios  of  lengths.  Other  curves 
(such  as  spirals,  quadratrixes,  and  cycloids)  I  call  geometrically  irrational. 
For  the  lengths  which  are  or  are  not  as  number  to  number  (according  to 
the  tenth  Book  of  Elements)  are  arithmetically  rational  or  irrational. 
And  therefore  I  cut  off  an  area  of  an  ellipsis  proportional  to  the  time  in 
which  it  is  described  by  a  curve  geometrically  irrational,  in  the  following 
manner. 

PROPOSITION  XXXI.     PROBLEM  XXIII. 

To  find  the  place  of  a  body  moving  in  a  given  elliptic  trajectory  at  any 

assigned  time. 

Suppose  A  to  be 
the  principal  vertex, 
S  the  focus,  and  O 
the  centre  of  the 
ellipsis  A  PB  ;  and 
let  P  be  the  place  of 
the  body  to  be  found. 
Produce  OA  to  G  so 
as  OG  may  be  to  OA 
as  OA  to  OS.  Erect 
the  perpendicular  GH;  and  about  the  centre  O,  with  the  interval  OG,  de 
scribe  the  circle*  GEF  ;  and  on  the  ruler  GH,  as  a  base,  suppose  the  wheel 
GEF  to  move  forwards,  revolving  about  its  axis,  and  in  the  mean  time  by 
its  point  A  describing  the  cycloid  ALL  Which  done,  take  GK  to  the 
perimeter  GEFG  of  the  wheel,  in  the  ratio  of  the  time  in  which  the  body 
proceeding  from  A  described  the  arc  AP,  to  the  time  of  a  whole  revolution 
in  the  ellipsis.  Erect  the  perpendicular  KL  meeting  the  cycloid  in  L ; 
then  LP  drawn  parallel  to  KG  will  meet  the  ellipsis  in  P,  the  required 
place  of  the  body. 

For  about  the  centre  O  with  the  interval  OA  describe  the  semi-circle 
AQB,  and  let  LP,  produced,  if  need  be,  meet  the  arc  AQ,  in  Q,  and  join 


SEC.  VI. 


OF    NATURAL    PHILOSOPHY. 


157 


SQ,  OQ.  Let  OQ  meet  the  arc  EFG  in  F,  and  upon  OQ  let  fall  the 
perpendicular  Sll.  The  area  APS  is  as  the  area  AQS,  that  is,  as  tlie 
difference  between  the  sector  OQA  and  the  triangle  OQS,  or  as  the  difLi- 
ence  of  the  rectangles  *OQ,  X  AQ,  and  -J.OQ  X  SR,  that  is,  because  ' .  >,_ 
is  given,  as  the  difference  between  the  arc  AQ  and  the  right  line  Sll :  ai.;l 
therefore  (because  of  the  equality  of  the  given  ratios  SR  to  the  sine  of  the 
arc  AQ,,  OS  to  OA,  OA  to  OG,  AQ  to  GF;  and  by  division,  AQ— Sii 
to  GF  —  sine  of  the  arc  AQ)  as  GK,  the  difference  between  the  arc  C 1 
and  tlie  sine  of  the  arc  AQ.  Q.E.D. 


SCHOLIUM. 

But  since  the  description  of  this  curve 
is  difficult,  a  solution  by  approximation 
will  be  preferable.  First,  then,  let  there 
be  found  a  certain  angle  B  which  may 
be  to  an  angle  of  57,29578  degrees, 
which  an  arc  equal  to  the  radius  subtends, 
as  SH,  the  distance  of  the  foci,  to  AB, 
the  diameter  of  the  ellipsis.  Secondly,  a  certain  length  L,  which  may  be  to 
the  radius  in  the  same  ratio  inversely.  And  these  being  found,  the  Problem 
may  be  solved  by  the  following  analysis.  By  any  construction  (or  even 
by  conjecture),  suppose  we  know  P  the  place  of  the  body  near  its  true 
place  jo.  Then  letting  fall  on  the  axis  of  the  ellipsis  the  ordinate  PR 
from  the  proportion  of  the  diameters  of  the  ellipsis,  the  ordinate  RQ  of 
the  circumscribed  circle  AQB  will  be  given  ;  which  ordinate  is  the  sine  of 
the  angle  AOQ,  supposing  AO  to  be  the  radius,  and  also  cuts  the  ellipsis 
in  P.  It  will  .be  sufficient  if  that  angle  is  found  by  a  rude  calculus  in 
numbers  near  the  truth.  Suppose  we  also  know  the  angle  proportional  to 
the  time,  that  is,  which  is  to  four  right  a  iules  as  the  time  in  which  tlie 
body  described  the  arc  A/?,  to  the  time  of  one  revolution  in  the  ellipsis. 
Let  this  angle  be  N.  Then  take  an  angle  D,  which  may  be  to  the  angle 
B  as  the  sine  of  the  angle  AOQ  to  the  radius ;  and  an  angle  E  which 
may  be  to  the  angle  N  —  AOQ  -fD  as  the  length  L  to  the  same  length 
L  diminished  by  the  cosine  of  the  angle  AOQ,  when  that  angle  is  less 
than  a  right  angle,  or  increased  thereby  when  greater.  In  the  next 
place,  take  an  angle  F  that  may  be  to  the  angle  B  as  the  sine  of  the  angle 
1OQ  H-  E  to  the  radius,  and  an  angle  G,  that  may  be  to  the  angle  N- 
AOQ  —  E  -f  F  as  the  length  L  to  the  same  length  L  diminished  by  the 
cosine  of  the  angle  AOQ  +  E,  when  that  angle  is  less  than  a  right  angle, 
or  increased  thereby  when  greater.  For  the  third  time  take  an  angle  H, 
that  may  be  to  the  angle  B  as  the  sine  of  the  angle  AOQ  f-  E  4-  G  to  the 
radius;  and  an  angle  I  to  the  angle  N — AOQ  —  E  —  G  -f-  H,  as  the 


58  THE   MATHEMATICAL    PRINCIPLES  jB(OK    1. 

length  L  is  to  the  same  length  L  diminished  by  the  cosine  of  the  angle 
AOQ  -f-  E  +  G,  when  that  angle  is  less  than  a  right  angle,  or  increased 
thereby  when  greater.  And  so  we  may  proceed  in  infinitum.  Lastly, 
take  the  angle  AOy  equal  to  the  angle  AOQ  -f-  E  4-  G  +  I  -\-}  &c.  and 
from  its  cosine  Or  and  the  ordinatejor,  which  is  to  its  sine  qr  as  the  lesser 
axis  of  the  ellipsis  to  the  greater,  \\  e  shall  have  p  the  correct  place  of  the 
body.  When  the  angle  N —  AOQ,  -f  D  happens  to  be  negative,  the 
sign  -|-  of  the  angle  E  must  be  every  where  changed  into  — ,  and  the  sign  — 
into  +.  And  the  same  thing  is  to  be  understood  of  the  signs  of  the  angles 
G  and  I,  when  the  angles  N  —  AOQ  —  E  -f  F,  and  N  —  AOQ  —  E  — 
G  +  H  come  out  negative.  But  the  infinite  series  AOQ  -f-  E  -f-  G  -|-  I  +, 
&c.  converges  so  very  fast,  that  it  will  be  scarcely  ever  needful  to  pro 
ceed  beyond  the  second  term  E.  And  the  calculus  is  founded  upon 
this  Theorem,  that  the  area  APS  is  as  the  difference  between  the  arc 
AQ  and  the  right  line  let  fall  from  the  focus  S  perpendicularly  upon  the 
radius  OQ. 

And  by  a  calculus  not  unlike,  the  Problem 
is  solved  in  the  hyperbola.  Let  its  centre  be 
O,  its  vertex  A,  its  focus  S,  and  asymptote 
OK ;  and  suppose  the  quantity  of  the  area  to 
be  cut  off  is  known,  as  being  proportional  to 
the  time.  Let  that  be  A,  and  by  conjecture 
suppose  we  know  the  position  of  a  rij;ht  i  ne 
SP,  that  cuts  off  an  area  APS  near  the  truth. 
Join  OP,  and  from  A  and  P  to  the  asymptote  °  T  A  S 

draw  AI,  PK  parallel  to  the  other  asymptote ;  and  by  the  table  of  loga 
rithms  the  area  AIKP  will  be  given,  and  equal  thereto  the  area  OPA, 
which  subducted  from  the  triangle  OPS,  will  leave  the  area  cut  off  APS. 
And  by  applying  2 APS  —  2 A,  or  2 A  —  2 A  PS,  the  double  difference  of 
the  area  A  that  was  to  be  cut  off,  and  the  area  APS  that  is  cut  off,  to  the 
line  SN  that  is  let  fall  from  the  focus  S,  perpendicular  upon  the  tangent 
TP,  we  shall  have  the  length  of  the  chord  PQ.  Which  chord  PQ  is  to 
be  inscribed  between  A  and  P,  if  the  area  APS  that  is  cut  off  be  greater 
than  the  area  A  that  was  to  be  cut  off,  but  towards  the  contrary  side  of  the 
point  P,  if  otherwise :  and  the  point  Q  will  be  the  place  of  the  body  more 
accurately.  And  by  repeating  the  computation  the  place  may  be  found 
perpetually  to  greater  and  greater  accuracy. 

And  by  such  computations  we  have  a  general 
analytical  resolution  of  the  Problem.  But  the  par 
ticular  calculus  that  follows  is  better  fitted  for  as 
tronomical  purposes.  Supposing  AO,  OB,  OD,  to 
be  the  semi-axis  of  the  ellipsis,  and  L  its  latus  rec 
tum,  and  D  the  difference  betwixt  the  lesser  semi- 


SEC.  VII.]  OF    NATURAL    PHILOSOPHY.  J  59 

axis  OD,  and  -,L  the  half  of  the  latus  rectum  :  let  an  angle  Y  be  found,  whose 
sine  may  be  to  the  radius  as  the  rectangle  under  that  difference  J),  and 
AO  4-  OD  the  half  sum  of  the  axes  to  the  square  of  the  greater  axis  AB. 
Find  also  an  angle  Z,  whose  sine  may  be  to  the  radius  as  the  double  rec 
tangle  under  the  distance  of  the  foci  SH  and  that  difference  D  to  triple 
the  square  of  half  the  greater  semi-axis  AO.  Those  angles  being  once 
found,  the  place  of  the  body  may  be  thus  determined.  Take  the  angle  T 
proportional  to  the  time  in  which  the  arc  BP  was  described,  or  equal  to 
what  is  called  the  mean  motion ;  and  an  angle  V  the  first  equation  of  thr 
mean  motion  to  the  angle  Y,  the  greatest  first  equation,  as  the  sine  of 
double  the  angle  T  is  to  the  radius ;  and  an  angle  X,  the  second  equation, 
to  the  angle  Z,  the  second  greatest  equation,  as  the  cube  of  the  sine  of  the 
angle  T  is  to  the  cube  of  the  radius.  Then  take  the  angle  BHP  the  mean 
motion  equated  equal  to  T  +  X  +  V,  the  sum  of  the  angles  T,  V.  X, 
if  the  angle  T  is  less  than  a  right  angle;  or  equal  to  T  +  X  —  V,  the 
difference  of  the  same,  if  that  angle  T  is  greater  than  one  and  less  than 
two  right  angles ;  and  if  HP  meets  the  ellipsis  in  P,  draw  SP,  and  it  will 
cut  off  the  area  BSP  nearly  proportional  to  the  time. 

This  practice  seems  to  be  expeditious  enough,  because  the  angles  V  and 
X,  taken  in  second  minutes,  if  you  please,  being  very  small,  it  will  be  suf 
ficient  to  find  two  or  three  of  their  first  figures.  But  it  is  likewise 
sufficiently  accurate  to  answer  to  the  theory  of  the  planet's  motions. 
For  even  in  the  orbit  of  Mars,  where  the  greatest  equation  of  the  centre 
amounts  to  ten  degrees,  the  error  will  scarcely  exceed  one  second.  But 
when  the  angle  of  the  mean  motion  equated  BHP  is  found,  the  angle  oi 
the  true  motion  BSP,  and  the  distance  SP,  are  readily  had  by  the  known 
methods. 

And  so  far  concerning  the  motion  of  bodies  in  curve  lines.  But  it  mav 
also  come  to  pass  that  a  moving  body  shall  ascend  or  descend  in  a  right 
line :  and  I  shall  now  go  on  to  explain  what  belongs  to  such  kind  of 
motions. 


SECTION  VII. 

Concerning  the  rectilinear  ascent  and  descent  of  bodies, 

PROPOSITION  XXXII.     PROBLEM  XXIV. 

Supposing  that  the  centripetal  force  is  reciprocally  proportional  to  tht 
square  of  tlie  distance  of  the  places  from  the  centre ;  it  is  required 
to  define  the  spaces  which  a  body,  falling  directly,  describes  in  given 
times. 
CASE  1.  If  the  body  does  not  fall  perpendicularly,  it  will  (by  Cor.  I 


160 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK  I 


Prop.  XIII)  describe  some  conic  section  whose  focus  is  A 
placed  in  the  centre  of  force.  Suppose  that  conic  sec 
tion  to  be  A  RPB  and  its  focus  S.  And,  first,  if  the 
figure  be  an  ellipsis,  upon  the  greater  axis  thereof  AB 
describe  the  semi-circle  ADB,  and  let  the  right  line 
I) PC  pass  through  the  falling  body,  making  right  angles 
with  the  axis;  and  drawing  DS,  PS,  the  area  ASD  will  c 
be  proportional  to  the  area  ASP,  and  therefore  also  to 
the  time.  The  axis  AB  still  reaiaining  the  same,  let  the 
breadth  of  the  ellipsis  be  perpetually  diminished,  and  s 
the  area  ASD  will  always  remain  proportional  to  the 
time.  Suppose  that  breadth  to  be  diminished  in,  in  fruit  um  ;  and  the  orbit 
APB  in  that  case  coinciding  with  the  axis  AB,  and  the  focus  S  with  the 
extreme  point  of  the  axis  B,  the  body  will  descend  in  the  right  line  AC1. 
and  the  area  ABD  will  become  proportional  to  the  time.  Wherefore  the 
space  AC  will  be  given  which  the  body  describes  in  a  given  time  by  its- 
perpendicular  fall  from  the  place  A,  if  the  area  ABD  is  taken  proportional 
to  the  time,  and  from  the  point  D  the  right  line  DC  is  let  fall  perpendic 
ularly  on  the  right  line  AB.  Q,.E.I. 

CASE  2.  If  the  figure  RPB  is  an  hyperbola,  on  the 
same  principal  diameter  AB  describe  the  rectangular 
hyperbola  BED  ;  and  because  the  areas  CSP,  CB/P, 
SPy  13,  are  severally  to  the  sev eral  areas  CSD,  CBED, 
SDEB,  in  the  given  ratio  of  the  heights  CP,  CD,  and 
the  area  SP/B  is  proportional  to  the  time  in  which 
the  body  P  will  move  through  the  arc  P/B.  the  area 
SDEB  will  be  also  proportional  to  that  time.  Let 
the  latus  rectum  of  the  hyperbola  RPB  be  diminished 
in  infitiitum,  the  latus  transversum  remaining  the 
same;  and  the  arc  PB  will  come  to  coincide  with  the 
right  line  CB,  and  the  focus  S,  with  the  vertex  B,  A- 
and  the  right  line  SD  with  the  right  line  BD.  And  therefore  the  area 
BDEB  will  be  proportional  to  the  time  in  which  the  body  C,  by  its  per 
pendicular  descent,  describes  the  line  CB.  Q.E.I. 

CASE  3.  And  by  the  like  argument,  if  the  figure 
RPB  is  a  parabola,  and  to  the  same  principal  ver 
tex  B  another  parabola  BED  is  described,  that 
may  always  remain  given  while  the  former  para 
bola  in  whose  perimeter  the  body  P  moves,  by 
having  its  latus  rectum  diminished  and  reduced 
to  nothing,  comes  to  coincide  with  the  line  CB, 
the  parabolic  segment  BDEB  will  be  proportional    if 
to  the  time  in  which  that  body  P  or  C  will  descend  to  the  centre  S  or  B 
Q.K.T 


fl.l 


OF    NATURAL    PHILOSOPHY. 


PROPOSITION  XXXIII.     THEOREM  IX. 

The  tilings  above  found  being  supposed.  I  say,  thai  the,  velocity  of  a  Jai 
ling  body  in  any  place  C  is  to  the  velocity  of  a  body,  describing  a 
circle  about  the  centre  B  at  the  distance  BC,  in,  the  subduplicate  ratio 
of  AC,  the  distance  of  the  body  from  the  remoter  vertex  A  of  the  circle 
or  rectangular  hyperbola,  to  iAB,  the  principal  semi-diameter  of  the 


Let  AB,  the  common  dia 
meter  of  both  figures  RPB, 
DEB,  be  bisected  in  O;  and 
draw  the  right  line  PT  that 
may  touch  the  figure  RPB 
in  P,  and  likewise  cut  that 
common  diameter  AB  (pro 
duced,  if  need  be)  in  T;  and 
let  SY  be  perpendicular  to 
this  line,  and  BQ  to  this  di 
ameter,  and  suppose  the  latus 
rectum  of  the  figure  RPB  to 
be  L.  Prom  Cor.  9,  Prop. 
XVI,  it  is  manifest  that  the 
velocity  of  a  body,  moving 
in  the  line  RPB  about  the 
centre  S,  in  any  place  P,  is 
to  the  velocity  of  a  body  describing  a  circle  about  the  same  centre,  at  the 
distance  SP,  in  the  subduplicate  ratio  of  the  rectangle  £L  X  SP  to  SY2 
Por  by  the  properties  of  the  conic  sections  ACB  is  to  CP2  as  2AO  to  L. 

2CP5  X  AO 

and  therefore  —  — rrrr; is  equal  to  L.     Therefore  those,  velocities  an 


o-- 


ACB 

to  each  other  in  the  subduplicate  ratio  of 


CP3  X  AO  X  SP 


ACB 


toSY~.  More 


over,  by  the  properties  of  the  conic  sections,  CO  is  to  BO  as  BO  to  TV.? 
and  (by  composition  or  division)  as  CB  to  BT.  Whence  (by  division  cs 
composition)  BO  —  or  +  CO  will  be  to  BO  as  CT  to  BT,  that  is,  AC 


CP2  X  AO  X  SP 

ACB" 


is  equal  to 


will  be  to  AO  as  CP  to  BQ;  and  therefore 

~AO  X  BC *     ^°W  suPPose  GV,  tne  breadth  of  the  figure  RPB,  to 

be  diminished  in  infinitum,  so  as  the  point  P  may  come  to  coincide  with 
the  point  C,  and  the  point  S  with  the  point  B.  and  the  line  SP  with  the 
line  BC,  and  the  line  SY  with  the  line  BQ;  and  the  velocity  of  the  body 
now  descending  perpendicularly  in  the  line  CB  will  be  to  the  velocity  of 

11 


162 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK     1 


a  body  describing  a  circle  about  the  centre  B,  at  the  distance  BC,  in  thr 

BQ2  X  AC  X  SP 

subduplicate  ratio  of  -  -r-^  -  ^  -  to  SY2,  that  is  (neglecting  the  ra- 


X  Jo 

tios  of  equality  of  SP  to  BC,  and  BQ,2  to  SY2),  in  the  subduplicate  ratio 
of  AC  to  AO,  or  iAB.  Q.E.D. 

COR.  1  .  When  the  points  B  and  S  come  to  coincide,  TC  will  become  to 
TS  as  AC  to  AO. 

COR.  2.  A  body  revolving  in  any  circle  at  a  given  distance  from  the 
centre,  by  its  motion  converted  upwards,  will  ascend  to  double  its  distance 

from  the  centre. 

• 

PROPOSITION  XXXIV.     THEOREM  X. 

If  the.  figure  BED  is  a  parabola,  I  say,  that  the  velocity  of  a  falling 
body  in  any  place  C  is  equal  to  the  velocity  by  which  a  body  may 
uniformly  describe  a  circle  about  the  centre  B  at  half  the  interval  BC 
For  (by  Cor.  7,  Prop.  XVI)  the  velocity  of  a 

body  describing  a  parabola  RPB  about  the  cen 

tre  S,  in  any  place  P,  is  equal  to  the  velocity  of 

a  body  uniformly  describing  a  circle  about  the  c 

same  centre  S  at  half  the  interval  SP.     Let  the 

breadth  CP  of  the  parabola  be  diminished  in 

itifiiiitirni,  so  as  the  parabolic  arc  P/B  may  come 

to  coincide  with  the  right  line  CB,  the  centre  S  s 

with  the  vertex  B,  and  the  interval  SP  with  the 

interval  BC,  and  the  proposition  will  be  manifest.     Q.E.D. 

PROPOSITION  XXXV.  THEOREM  XL 

The  same  things  supposed,  I  say,  that  the  area  of  the  figure  DES,  de 
scribed  by  the  indefinite  radius  SD,  is  equal  to  the  area  which  a  body 
with  a  radius  equal  to  h'df  the  latus  rectum  of  the  figure  DES,  by 
uniformly  revolving  about  the  centre  S,  may  describe  in  the  same  tijiw. 


1 


JD/ 


AJ 


SEC.  ni: 


OF  NATURAL    PHILOSOPHY. 


For  suppose  a  body  C  in  the  smallest  moment  of  time  describes  in  fal 
ling  the  infinitely  little  line  Cc.  while  another  body  K,  uniformly  revolv 
ing  about  the  centre  S  in  the  circle  OK/r,  describes  the  arc  KA:.  Erect  the 
perpendiculars  CD,  cd,  meeting  the  figure  DES  in  D,  d.  Join  SD,  Sf/. 
SK.  SA*;  and  draw  Del  meeting  the  axis  AS  in  T,  and  thereon  let  fall  the 
perpendicular  SY. 

CASE  1.  If  the  figure  DES  is  a  circle,  or  a  rectangular  hyperbola,  bisect 
its  transverse  diameter  AS  in  O,  and  SO  will  be  half  the  latus  rectum. 
And  because  TC  is  to  TD  as  Cc  to  Dd,  and  TD  to  TS  as  CD  to  S  Y ; 
ex  aquo  TC  will  be  to  TS  as  CD  X  Cc  to  SY  X  Dd.  But  (by  Cor.  1, 
Prop.  XXXIII)  TC  is  to  TS  as  AC  to  AO;  to  wit,  if  in  the  coalescence 
of  the  points  D,  d,  the  ultimate  ratios  of  the  lines  are  taken.  Wherefore 
AC  is  to  AO  or  SK  as  CD  X  Cc  to  S  Y  X  Vd,  Farther,  the  velocity  of 
the  descending  body  in  C  IF,  to  the  velocity  of  a  body  describing  a  circle 
about  the  centre  S,  at  the  interval  SC,  in  the  subduplicate  ratio  of  AC  to 
AO  or  SK  (by  Pi-op.  XXXIII) ;  and  this  velocity  is  to  the  velocity  of  a 
body  describing  the  circle  OKA:  in  the  subduplicate  ratio  of  SK  to  SC 
(by  Cor.  6,  Prop  IV) ;  and,  ex  aqnnj  the  first  velocity  to  the  last,  that  is, 
the  little  line  Cc  to  the  arc  K/r,  in  the  subduplicate  ratio  of  AC  to  SC, 
that  is,  in  the  ratio  of  AC  to  CD.  Wherefore  CD  X  Cc  is  equal  to  AC 
X  KA*,  and  consequently  AC  to  SK  as  AC  X  KA:  to  SY  X  IW.  and 
thence  SK  X  KA:  equal  to  SY  X  Drf,  and  iSK  X  KA:  equal  to  £SY  X  DC/, 
that  is,  the  area  KSA*  equal  to  the  area  SDrf.  Therefore  in  every  moment 
of  time  two  equal  particles,  KSA"  and  SDrf,  of  areas  are  generated,  which, 
if  their  magnitude  is  diminished,  and  their  number  increased  in  iiifinif't-w, 
obtain  the  ratio  cf  equality,  and  consequently  (by  Cor.  Lem.  IV),  the  whole 
areas  together  generated  are  always  equal.  Q..E.D. 

CASE  2.  But  if  the  figure  DES  is  a 
parabola,  we  shall  find,  as  above.  CD  X 
Cc  to  SY  X  Df/  as  TC  to  TS,  that  is, 
as  2  to  1  ;  and  that  therefore  |CD  X  Cc 
is  equal  to  i  SY  X  Vd.  But  the  veloc 
ity  of  the  falling  body  in  C  is  equal  to 
the  velocity  writh  which  a  circle  may  be 
uniformly  described  at  the  interval  4SC 
(by  Prop"  XXXIV).  And  this  velocity 
to  the  velocity  with  which  a  circle  may 
be  described  with  the  radius  SK,  that  is, 
the  little  line  Cc  to  the  arc  KA',  is  (by 
Cor.  6,  Prop.  IV)  in  the  subduplicate  ratio  of  SK  to  iSC ;  that  is,  in  the 
ratio  of  SK  to  *CD.  Wherefore  iSK  X  KA:  is  equal  to  4CD  X  Cc,  and 
therefore  equal  to  £SY  X  T)d  ;  that  is,  the  area  KSA*  is  equal  to  the  area 
SIW,  as  above.  Q.E.D. 


164 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK    1. 


PROPOSITION  XXXVI.     PROBLEM  XXV. 

To  determine  the  times  of  the  descent  of  a  body  falling  from 

place  A. 

Upon  the  diameter  AS,  the  distance  of  the  body  from  the 
centre  at  the  beginning,  describe  the  semi-circle  ADS,  as 
likewise  the  semi-circle  OKH  equal  thereto,  about  the  centre 
S.  From  any  place  C  of  the  body  erect  the  ordinate  CD.  ° 
Join  SD,  and  make  the  sector  OSK  equal  to  the  area  ASD. 
It  is  evident  (by  Prop.  XXXV)  that  the  body  in  falling  will 
describe  the  space  AC  in  the  same  time  in  which  another  body, 
uniformly  revolving  about  the  centre  S,  may  describe  the  arc 
OK.  Q.E.F.  M 


a  given 


PROPOSITION  XXXVII.     PROBLEM  XXVI. 

To  define  the  times  of  the  ascent  or  descent  of  a  body  projected  upwards 

or  downwards  from  a  given  place. 

Suppose  the  body  to  go  oif  from  the  given  place  G,  in  the  direction  of 
the  line  GS,  with  any  velocity.  In  the  duplicate  ratio  of  this  velocity  to 
the  uniform  velocity  in  a  circle,  with  which  the  body  may  revolve  about 


\ 


H 


D 


the  centre  S  at  the  given  interval  SG,  take  GA  to  £AS.  If  that  ratio  is 
the  same  as  of  the  number  2  to  1,  the  point  A  is  infinitely  remote ;  in 
which  case  a  parabola  is  to  be  described  with  any  latus  rectum  to  the  ver 
tex  S,  and  axis  SG  ;  as  appears  by  Prop.  XXXIV.  But  if  that  ratio  is 
less  or  greater  than  the  ratio  of  2  to  1,  in  the  former  case  a  circle,  in  the 
latter  a  rectangular  hyperbola,  is  to  be  described  on  the  diameter  SA;  as 
appears  by  Prop.  XXXIII.  Then  about  the  centre  S,  with  an  interval 
equal  to  half  the  latus  rectum,  describe  the  circle  H/vK ;  and  at  the  place 
G  of  the  ascending  or  descending  body,  and  at  any  other  place  C,  erect  the 
perpendiculars  GI,  CD,  meeting  the  conic  section  or  circle  in  I  and  D. 
Then  joining  SI,  SD,  let  the  sectors  HSK,  HS&  be  made  equal  to  the 
segments  SEIS,  SEDS.  and  (by  Prop.  XXXV)  the  body  G  will  describe 


SEC.  VII.] 


OF    NATURAL    PHILOSOPHY. 


165 


the  space  GO  in  the  same  time  in  which  the  body  K  may  describe  t*he  arc 
Kk.     Q.E.F. 

PROPOSITION  XXXVIII.     THEOREM  XII. 

Supposing  that  the  centripetal  force  is  proportional  to  the  altitude  or 
distance  of  places  from  the  centre,  I  say,  that  the  times  and  velocities 
of  falling  bodies,  and  the  spaces  which  they  describe,  are  respectively 
proportional  to  the  arcs,  and  the  right  and  versed  sines  of  the  arcs. 
Suppose  the  body  to  fall  from  any  place  A  in  the     A. 
right  line  AS ;  and  about  the  centre  of  force  S,  with 
the  interval  AS,  describe  the  quadrant  of  a  circle  AE ; 
and  let  CD  be  the  right  sine  of  any  arc  AD ;  and  the 
body  A  will  in  the  time  AD   in  falling  describe  the 
space  AC,  and  in  the  place  C   will  acquire  the  ve 
locity  CD. 

This  is  demonstrated  the  same  way  from  Prop.  X,  as  Prop.  XXX11  was 
demonstrated  from  Prop.  XI. 

COR.  1.  Hence  the  times  are  equal  in  which  one  body  falling  from  the 
place  A  arrives  at  the  centre  S,  and  another  body  revolving  describes  the 
quadrantal  arc  ADE. 

COR.  2.  Wherefore  all  the  times  are  equal  in  which  bodies  falling  from 
whatsoever  places  arrive  at  the  centre.  For  all  the  periodic  times  of  re 
volving  bodies  are  equal  (by  Cor.  3;  Prop.  IV). 

PROPOSITION  XXXIX.     PROBLEM   XXVIT. 

Supposing  a  centripetal  force  of  any  kind,  and  granting  the  quadra- 
tnres  of  curvilinear  figures  ;  it  is  required  to  find  the  velocity  of  a  bod)/, 
ascending  or  descending  in  a  right  line,  in  the  several  places  through 
which  it  passes  ;  as  also  the  time  in  which  it  will  arrive  at  any  place : 
and  vice  versa. 
Suppose  the  body  E  to  fall  from  any  place 

A  in  the  right  line  AD  EC  ;  and  from  its  place 

E  imagine  a  perpendicular  EG  always  erected 

proportional  to  the  centripetal  force  in   that 

place  tending  to  the  centre  C ;  and  let  BFG 

be  a  curve  line,  the  locus  of  the  point  G.     And  D 

in  the  beginning  of  the  motion  suppose  EG  to 

coincide  with  the  perpendicular  AB ;  and  the 

velocity  of  the  body  in  any  place  E  will  be  as 

a  right  line  whose  square  is  equal  to  the  cur 
vilinear  area  ABGE.     Q.E.I. 
In  EG  take  EM  reciprocally  proportional  to 


E 


366  THE    MATHEMATICAL    PRINCIPLES  [BOOK    1 

a  right  line  whose  square  is  equal  to  the  area  ABGE,  and  let  VLM  be  a 
curve  line  wherein  the  point  M  is  always  placed,  and  to  which  the  right 
line  AB  produced  is  an  asymptote;  and  the  time  in  which  the  body  in 
falling-  describes  the  line  AE,  will  be  as  the  curvilinear  area  ABTVME. 
Q.E.I. 

For  in  the  right  line  AE  let  there  be  taken  the  very  small  line  DE  of 
a  given  length,  and  let  DLF  be  the  place  of  the  line  EMG,  when  the 
body  was  in  D  ;  and  if  the  centripetal  force  be  such,  that  a  right  line, 
whose  square  is  equal  to  the  area  ABGE;  is  as  the  velocity  of  the  descend 
ing  body,  the  area  itself  will  be  as  the  square  of  that  velocity  ;  that  is,  if 
for  the  velocities  in  D  and  E  we  write  V  and  V  +  I,  the  area  ABFD  will 
be  as  VY,  and  the  area  ABGE  as  YY  +  2VI  -f  II;  and  by  division,  the 

area  DFGE  as  2  VI  -f  LI,  and  therefore     ^      will  be  as  --  ^r 


that  is.  if  we  take  the  first  ratios  of  those  quantities  when  just  nascent,  the 

2YI 

length  DF  is  as  the  quantity  -|yrr»  an(i  therefore  also  as  half  that  quantity 

1  X  Y 

•     But  the  time  in  which  the  body  in  falling  describes  the  very 


line  DE,  is  as  that  line  directly  and  the  velocity  Y  inversely  ;  and 
the  force  will  be  as  the  increment  I  of  the  velocity  directly  and  the  time 
inversely  ;  and  therefore  if  we  take  the  first  ratios  when  those  quantities 

I  X  V 

are  just  nascent,  as—  -jy==r-.  that  is,  as  the  length  DF.     Therefore  a  force 

proportional  to  DF  or  EG  will  cause  the  body  to  descend  with  a  velocity 
that  is  as  the  right  line  whose  square  is  equal  to  the  area  ABGE.  Q.E.D. 

Moreover,  since  the  time  in  which  a  very  small  line  DE  of  a  given 
length  may  be  described  is  as  the  velocity  inversely,  and  therefore  also 
inversely  as  a  right  line  whose  square  is  equal  to  the  area  ABFD  ;  and 
since  the  line  DL.  and  by  consequence  the  nascent  area  DLME,  will  be  as 
(he  same  right  line  inversely,  the  time  will  be  as  the  area  DLME,  and 
the  sum  of  all  the  times  will  be  as  the  sum  of  all  the  areas  :  that  is  (by 
Cor.  Lern.  IV),  the  whole  time  in  which  the  line  AE  is  described  will  be 
as  the  whole  area  ATYME.  Q.E.D. 

COR.  1.  Let  P  be  the  place  from  whence  a  body  ought  to  fall,  so  as 
that,  when  urged  by  any  known  uniform  centripetal  force  (such  as 
gravity  is  vulgarly  supposed  to  be),  it  may  acquire  in  the  place  D  a 
velocity  equal  to  the  velocity  which  another  body,  falling  by  any  force 
whatever,  hath  acquired  in  that  place  D.  In  the  perpendicular  DF  let 
there  be  taken  DR.,  which  may  be  o  DF  as  that  uniform  force  to 
the  other  force  in  the  place  D.  Complete  the  rectangle  PDRQ,,  and  cut 
iff  the  area.  ABFD  equal  to  that  rectangle.  Then  A  will  be  the  place 


SEC.  VII.  I 


OF    NATURAL    PHILOSOPHY. 


10; 


from  whence  the  other  body  fell.  For  com 
pleting  the  rectangle  DRSE,  since  the  area 
ABFD  is  to  the  area  DFGE  as  VV  to  2VI, 
and  therefore  as  4V  to  I,  that  is,  as  half  the 
whole  velocity  to  the  increment  of  the  velocity 
of  the  body  falling  by  the  unequable  force ;  and 
in  like  manner  the  area  PQRD  to  the  area 
DRSE  as  half  the  whole  velocity  to  the  incre 
ment  of  the  velocity  of  the  body  falling  by  the 
uniform  force ;  and  since  those  increments  (by 
reason  of  the  equality  of  the  nascent  times) 
are  as  the  generating  forces,  that  is,  as  the  or- 
dinates  DF,  DR,  and  consequently  as  the  nascent  areas  DFGE,  DRSE : 
therefore,  ex  aq-uo,  the  whole  areas  ABFD,  PQRD  will  be  to  one  another 
as  the  halves  of  the  whole  velocities ;  and  therefore,  because  the  velocities 
are  equal,  they  become  equal  also. 

COR.  2.  Whence  if  any  body  be  projected  either  upwards  or  downwards 
with  a  given  velocity  from  any  place  D,  and  there  be  given  the  law  of 
centripetal  force  acting  on  it,  its  velocity  will  be  found  in  any  other  place, 
as  e,  by  erecting  the  ordinate  eg,  and  taking  that  velocity  to  the  velocity 
in  the  place  D  as  a  right  line  whose  square  is  equal  to  the  rectangle 
PQRD,  either  increased  by  the  curvilinear  area  DFge,  if  the  place  e  is 
below  the  place  D,  or  diminished  by  the  same  area  DFg-e,  if  it  be  higher, 
is  to  the  right  line  whose  square  is  equal  to  the  rectangle  PQRD  alone. 

COR.  3.  The  time  is  also  known  by  erecting  the  ordinate  em  recipro 
cally  proportional  to  the  square  root  of  PQRD  -f-  or  —  T)Fge,  and  taking 
the  time  in  which  the  body  has  described  the  line  De  to  the  time  in  which 
another  body  has  fallen  with  an  uniform  force  from  P,  and  in  falling  ar 
rived  at  D  in  the  proportion  of  the  curvilinear  area  DLme  to  the  rectan 
gle  2PD  X  DL.  For  the  time  in  which  a  body  falling  with  an  uniform 
force  hath  described  the  line  PD,  is  to  the  time  in  which  the  same  body 
has  described  the  line  PE  in  the  subduplicate  ratio  of  PD  to  PE ;  that  is 
(the  very  small  line  DE  being  just  nascent),  in  the  ratio  of  PD  to  PD  -f 
^DE;  or  2PD  to  2PD  -f-  DE,  and,  by  division,  to  the  time  in  which  the 
body  hath  described  the  small  line  DE,  as  2PD  to  DE,  and  therefore  as 
the  rectangle  2PD  X  DL  to  the  area  DLME ;  and  the  time  in  which 
both  the  bodies  described  the  very  small  line  DE  is  to  the  time  in  which 
the  body  moving  unequably  hath  described  the  line  De  as  the  area  DLME 
to  the  area  DLme  ;  and,  ex  aquo,  the  first  mentioned  of  these  times  is  to 
the  last  as  the  rectangle  2PD  X  DL  to  the  area  DLrae. 


163  THE    MATHEMATICAL    PRINCIPLES  [BoOK    I 

SECTION  VIII. 

Of  the  invention  of  orbits  wherein  bodies  will  revolve,  being  acted  upon 
by  any  sort  of  centripetal  force. 

PROPOSITION  XL.     THEOREM  XIII. 

//'  a  body,  acted  upon  by  any  centripetal  force,  is  any  how  moved,  and 
another  body  ascends  or  descends  in  a  right  line,  and  their  velocities 
be  equal  in  amj  one  case  of  equal  altitudes,  t/ieir  velocities  will  be  also 
equal  at  all  equal  altitudes. 

Let  a  body  descend  from  A  through  D  and  E,  to  the  centre 
(j :  and  let  another  body  move  from  V  in  the  curve  line  VIK&. 
From  the  centre  C,  with  any  distances,  describe  the  concentric 
circles  DI,  EK,  meeting  the  right  line  AC  in  I)  and  E;  and 
the  curve  VIK  in  I  and  K.  Draw  1C  meeting  KE  in  N,  and 
on  IK  let  fall  the  perpendicular  NT  •  and  let  the  interval  DE 
or  IN  between  the  circumferences  of  the  circles  be  very  small ;  K/ 
and  imagine  the  bodies  in  D  and  I  to  have  equal  velocities. 
Then  because  the  distances  CD  and  CI  are  equal,  the  centri 
petal  forces  in  D  and  I  will  be  also  equal.  Let  those  forces  be  k) 
expressed  by  the  equal  lineoke  DE  and  IN ;  and  let  the  force 
IN  (by  Cor.  2  of  the  Laws  of  Motion)  be  resolved  into  two 
others,  NT  and  IT.  rl  hen  the  force  NT  acting  in  the  direction 
line  NT  perpendicular  to  the  path  ITK  of  the  body  will  not  at  all  affect 
or  change  the  velocity  of  the  body  in  that  path,  but  only  draw  it  aside 
from  a  rectilinear  course,  and  make  it  deflect  perpetually  from  the  tangent 
of  the  orbit,  and  proceed  in  the  curvilinear  path  ITK/j.  That  whole 
force,  therefore,  will  be  spent  in  producing  this  effect:  but  the  other  force 
IT,  acting  in  the  direction  of  the  course  of  the  body,  will  be  all  employed 
in  accelerating  it,  and  in  the  least  given  time  will  produce  an  acceleration 
proportional  to  itself.  Therefore  the  accelerations  of  the  bodies  in  D  and 
I,  produced  in  equal  times,  are  as  the  lines  DE,  IT  (if  we  take  the  first 
ratios  of  the  nascent  lines  DE,  IN,  IK,  IT,  NT) ;  and  in  unequal  times  as 
those  lines  and  the  times  conjunctly.  But  the  times  in  which  DE  and  IK 
are  described,  are,  by  reason  of  the  equal  velocities  (in  D  and  I)  as  the 
spaces  described  DE  and  IK,  and  therefore  the  accelerations  in  the  course 
of  the  bodies  through  the  lines  DE  and  IK  are  as  DE  and  IT,  and  DE 
and  IK  conjunctly ;  that  is,  as  the  square  of  DE  to  the  rectangle  IT  into 
IK.  But  the  rectangle  IT  X  IK  is  equal  to  the  square  of  IN,  that  is, 
equal  to  the  square  of  DE ;  and  therefore  the  accelerations  generated  in 
the  passage  of  the  bodies  from  D  and  I  to  E  and  K  are  equal.  Therefore 
the  velocities  of  the  holies  in  E  and  K  are  also  equal,  and  by  the  same 
reasoning  they  will  always  be  found  equal  in  any  subsequent  equal  dis 
tances.  Q..E.D. 


SEC.  VI11.J  OF    NATURAL    PHILOSOPHY.  169 

By  the  same  reasoning,  bodies  of  equal  velocities  and  equal  distances 
from  the  centre  will  be  equally  retarded  in  their  ascent  to  equal  distances. 
Q.E.D. 

COR.  1.  Therefore  if  a  body  either  oscillates  by  hanging  to  a  string,  or 
by  any  polished  and  perfectly  smooth  impediment  is  forced  to  move  in  a 
curve  line  ;  and  another  body  ascends  or  descends  in  a  right  line,  and  their 
velocities  be  equal  at  any  one  equal  altitude,  their  velocities  will  be  also 
equal  at  all  other  equal  altitudes.  For  by  the  string  of  the  pendulous 
body,  or  by  the  impediment  of  a  vessel  perfectly  smooth,  the  same  thing 
will  be  effected  as  by  the  transverse  force  NT.  The  body  is  neither 
accelerated  nor  retarded  by  it,  but  only  is  obliged  to  leave  its  rectilinear 
course. 

COR.  2.  Suppose  the  quantity  P  to  be  the  greatest  distance  from  the 
centre  to  which  a  body  can  ascend,  whether  it  be  oscillating,  or  revolving 
in  a  trajectory,  and  so  the  same  projected  upwards  from  any  point  of  a 
trajectory  with  the  velocity  it  has  in  that  point.  Let  the  quantity  A  be 
the  distance  of  the  body  from  the  centre  in  any  other  point  of  the  orbit ;  and 
let  the  centripetal  force  be  always  as  the  power  An — ',  of  the  quantity  A,  the 
index  of  which  power  n  —  1  is  any  number  n  diminished  by  unity.  Then 
the  velocity  in  every  altitude  A  will  be  as  v/  P11  —  A",  and  therefore  will 
be  given.  For  by  Prop.  XXXIX,  the  velocity  of  a  body  ascending  and 
descending  in  a  right  line  is  in  tha't  very  ratio. 

PROPOSITION  XLI.     PROBLEM  XXVTII. 

Supposing  a  centripetal  force  of  any  kind,  and  granting  the  quadra 
tures  of  curvilinear  figures,  it  is  required  to  find  as  well  the  trajecto 
ries  in  which  bodies  will  move,  as  the  times  of  their  motions  in  the 
trajectories  found. 
Let  any  centripetal  force  tend  to 
the  centre  C,  and  let  it  be  required 
to  find   the   trajectory  VIKAr.      Let  R, 
there  be  given  the  circle  VR,  described 
from  the  centre  C  with  any  interval 
CV;  and  from  the  same  centre  de 
scribe  any  other  circles  ID,  KE  cut 
ting  the  trajectory  in  I  and  K,  and 
the  right  line  CV  in  D  and  E.  Then 
draw  the  right  line  CNIX  cutting  the  c 

circles  KE,  VR  in  N  and  X,  and  the  right  line  CKY  meeting  the  circle 
VJi  in  Y.  Let  the  points  I  and  K  be  indefinitely  near ;  and  let  the  body 
go  on  from  V  through  I  and  K  to  k  ;  and  let  the  point  A  be  the  place 
from  whence  anothe  body  is  to  fall,  so  as  in  the  place  D  to  acquire  a  ve 
locity  equal  to  the  velocity  of  the  first  body  in  I.  And  things  remaining 
as  in  Prop.  XXXIX,  the  lineola  IK,  described  in  the  least  given  time 


THE    MATHEMATICAL    PRINCIPLES  [BOOK    1 

trill  be  as  the  velocity,  and  therefore  as  the  right  line  whose  square  is 
equal  to  the  area  ABFD,  and  the  triangle  ICK  proportional  to  the  time 
will  be  given,  and  therefore  KN  will  be  reciprocally  as  the  altitude  1C  : 
that  is  (if  there  be  given  any  quantity  Q,  and  the  altitude  1C  be  called 

A),  as  -T-.     This  quantity  —  call  Z,  and  suppose  the  magnitude  of  Q,  to 

oe  such  that  in  some  case  v/ABFD  may  be  to  Z  as  IK  to  KN,  and  then 
in  all  cases  V  ABFD  will  be  to  Z  as  IK  to  KN,  and  ABFD  to  ZZ  as 
IK2  to  KN2,  and  by  division  ABFD  —  ZZ  to  ZZ  as  IN2  to  KN2,  and  there- 


fore   V  ABFD  —  ZZ  to  Z,  or  —  as  IN  to  KN;  and  therefore  A  x  KN 

Q.  x  IN 

\vill  be  equal  to  —  —  —  .  Therefore  since  YX  X  XC  is  to  A  X  KN 

—  ZZ 


Q.  X  IN  x  CX2 
as  CX2,  to  AA,  the  rectangle  XY  X  XC  will  be  equal  to- 


AAv/ABFD  — ZZ. 

Therefore  in  the  perpendicular  DF  let  there  be  taken  continually  I)//,  IV 

Q  ax  ex2 

equal  to — ,  =. —         respectively,    and 

2  v/  ABFD  —  ZZ    2AA  V  ABFD  —  ZZ 

let  the  curve  lines  ab,  ac,  the  foci  of  the  points  b  and  c,  be  described  :  and 
from  the  point  V  let  the  perpendicular  Va  be  erected  to  the  line  AC,  cut 
ting  off  the  curvilinear  areas  VD&a,  VDra,  and  let  the  ordi nates  Es:? 
E#,  be  erected  also.  Then  because  the  rectangle  D&  X  IN  or  DbzR  is 
equal  to  half  the  rectangle  A  X  KN,  or  to  the  triangle  ICK  ;  and  the 
rectangle  DC  X  IN  or  Dc.rE  is  equal  to  half  the  rectangle  YX  X  XC,  or 
to  the  triangle  XCY;  that  is,  because  the  nascent  particles  I)6d3,  ICK 
of  the  areas  VD/>#,  VIC  are  always  equal;  and  the  nascent  particles 
Dc^-E,  XCY  of  the  areas  VDca,  VCX  are  always  equal :  therefore  the 
generated  area  VD6a  will  be  equal  to  the  generated  area  VIC,  and  there 
fore  proportional  to  the  time;  and  the  generated  area  VDco-  is  equal  to 
the  generated  sector  VCX.  If,  therefore,  any  time  be  given  during  which 
the  body  has  been  moving  from  V,  there  will  be  also  given  the  area  pro 
portional  to  it  VD/>«;  and  thence  will  be  given  the  altitude  of  the  body 
CD  or  CI ;  and  the  area  VDca,  and  the  sector  VCX  equal  there' o,  together 
with  its  angle  VCL  But  the  angb  VCI,  and  the  altitude  CI  being  given, 
there  is  also  given  the  place  I,  in  which  the  body  will  be  found  at  the  end 
of  that  time.  Q.E.I. 

COR.  1.  Hence  the  greatest  and  least  altitudes  of  the  bodies,  that  is,  the 
apsides  of  the  trajectories,  may  be  found  very  readily.  For  the  apsides 
are  those  points  in  which  a  right  line  1C  drawn  through  the  centre  falls 
perpendicularly  upon  the  trajectory  VTK;  which  comes  to  pass  when  the 
right  lines  IK  and  NK  become  equal;  that  is,  when  the  area  ABFD  ig 
C'nl  to  ZZ. 


OF  NATURAL    PHILOSOPHY. 


171 


SEC.  VI1LJ 

COR.  2.  So   also   the  angle  KIN,  in  which  the  trajectory  at  any  place 
cuts  the  line  1C.  may  be  readily  found  by  the  given  altitude  1C  of  the 
body :  to  wit,  by  making  the  sine  of  that  angle  to  radius  as  KN  to  IK 
that  is,  as  Z  to  the  square  root  of  the  area  ABFD. 

COR.  3.  If  to  the  centre  C,  and  the 
principal  vertex  V,  there  be  described  a 
conic  section  VRS ;  and  from  any  point 
thereof,  as  R,  there  be  drawn  the  tangent  T 
RT  meeting  the  axis  CV  indefinitely  pro 
duced  in  the  point  T ;  and  then  joining  C 
CR  there  be  drawn  the  right  line  CP,  Q- 
equal  to  the  abscissa  CT,  making  an  angle  VCP  proportional  to  the  sector 
VCR ;  and  if  a  centripetal  force,  reciprocally  proportional  to  the  cubes 
of  the  distances  of  the  places  from  the  centre,  tends  to  the  centre  C ;  and 
from  the  place  V  there  sets  out  a  body  with  a  just  velocity  in  the  direc 
tion  of  a  line  perpendicular  to  the  right, line  CV;  that  body  will  proceed 
in  a  trajectory  VPQ,,  which  the  point  P  will  always  touch ;  and  therefore 
if  the  conic  section  VI\  S  be  an  hyberbola,  the  body  will  descend  to  the  cen 
tre  ;  but  if  it  be  an  ellipsis,  it  will  ascend  perpetually,  and  go  farther  and 
farther  off  in  infinilum.  And,  on  the  contrary,  if  a  body  endued  with  any 
velocity  goes  off  from  the  place  V,  and  according  as  it  begins  either  to  de 
scend  obliquely  to  the  centre,  or  ascends  obliquely  from  it,  the  figure  VRS 
be  either  an  hyperbola  or  an  ellipsis,  the  trajectory  may  be  found  by  increas 
ing  or  diminishing  the  angle  VCP  in  a  given  ratio.  And  the  centripetal 
force  becoming  centrifugal,  the  body  will  ascend  obliquely  in  the  trajectory 
VPQ,  which  is  found  by  taking  the  angle  VCP  proportional  to  the  elliptic 
sector  VRC,  and  the  length  CP  equal  to  the  length  CT,  as  before.  All  these 
things  follow  from  the  foregoing  Proposition,  by  the  quadrature  of  a  certain 
ourve,  the  invention  of  which,  as  being  easy  enough,  for  brevity's  sake  I  omit. 

PROPOSITION  XLII.     PROBLEM  XXIX. 

The  law  of  centripetal  force  being  given,  it  is  required  to  find  the  motion 
of  a  body  setting  out  from  a  given  place,  with  a  given  velocity,  in  the 
direction  of  a  given  right  line. 
Suppose  the  same  things  as  in 

Ihe  three  preceding  propositions; 

and    let    the    body    go    off    from 

the  place  I  in  the  direction  of  the 

little  line,  IK,  with  the  same  ve 
locity  as  another  body,  by  falling 

with  an  uniform  centripetal  force 

from  the  place  P,  may  acquire  in 

I);  and  let  this  uniform  force  be 

to  the  force  with  which  the  body 


1.72  THE   MATHEMATICAL    PRINCIPLES  [BOOK    1. 

is  at  first  urged  in  I,  as  DR  to  DF.  Let  the  body  go  on  towards  k;  and 
about  the  centre  C,  with  the  interval  Ck,  describe  the  circle  ke,  meeting 
the  right  line  PD  in  e,  and  let  there  be  erected  the  lines  eg,  ev,  ew,  ordi- 
nately  applied  to  the  curves  BF§*,  abv}  acw.  From  the  given  rectangle 
PDRQ,  and  the  given  law  of  centripetal  force,  by  which  the  first  body  is 
acted  on,  the  curve  line  BF»*  is  also  given,  by  the  construction  of  Prop. 
XXVII,  and  its  Cor.  1.  Then  from  the  given  angle  CIK  is  given  the 
proportion  of  the  nascent  lines  1K;  KN ;  and  thence,  by  the  construction 
of  Prob.  XXVIII,  there  is  given  the  quantity  Q,,  with  the  curve  lines  abv, 
acw ;  and  therefore,  at  the  end  of  any  time  Dbve,  there  is  given  both 
the  altitude  of  the  body  Ce  or  Ck,  and  the  area  Dcwe,  with  the  sector 
equal  to  it  XCy,  the  angle  1C  A:,  and  the  place  k}  in  which  the  body  will 
then  be  found.  Q.E.I. 

We  suppose  in  these  Propositions  the  centripetal  force  to  vary  in  its 
recess  from  the  centre  according  to  some  law,  which  any  one  may  imagine 
at  pleasure;  but  at  equal  distances  from  the  centre  to  be  everywhere  the 
Bame. 

I  have  hitherto  considered  the  motions  of  bodies  in  immovable  orbits. 
It  remains  now  to  add  something  concerning  their  motions  in  orbits  which 
revolve  round  the  centres  of  force. 


SECTION  IX. 

Of  the  motion  of  bodies  in  moveable  orbits  ;  and  of  the  motion  of  the 

apsides. 

PROPOSITION  XLIII.    PROBLEM   XXX. 

Ft  is  required  to  make  a  body  move  in  a  trajectory  that  revolves  about 
the  centre  of  force  in  the  same  manner  as  another  body  in  the  same 
trajectory  at  rest. 
In.  the  orbit  VPK,  given  by  position,  let  the  body 

P  revolve,   proceeding  from   V  towards  K.     From 

the  centre  C  let  there  be  continually  drawn  Cp,  equal 

to  CP,  making  the  angle  VC/?  proportional  to  the 

angle  VCP ;  and  the  area  which  the  line  Cp  describes 

will  be  to  the  area  VCP,  which  the  line  CP  describes 

at  the  same  time,  ns  the  velocity  of  the  describing 

line  Cp  to  the  velocity  of  the  describing  line  CP ; 

that  is,  as  the  angle  VC/?  to  the  angle  VCP,  therefore  in  a  given  ratio, 

and  therefore  proportional  to  the  time.     Since,  then,  the  area  described  by 

the  line  Cp  in  an  immovable  plane  is  proportional  to  the  time,  it  is  manifest 

that  a  body,  being  acted  upon  by  a  just  quantity  of  centripetal  force  may 


SEC.  L\.] 


OF    NATURAL    PHILOSOPHY. 


173 


revolve  with  the  point  p  in  the  curve  line  which  the  same  point  p,  by  the 
method  just  now  explained,  may  be  made  to  describe  an  immovable  plane. 
Make  the  angle  VC^  equal  to  the  angle  PC/?,  and  the  line  Cu  equal  to 
CV,  and  the  figure  uCp  equal  to  the  figure  VCP;  and  the  body  being  al 
ways  in  the  point  p}  will  move  in  the  perimeter  of  the  revolving  figure 
nCp,  and  will  describe  its  (revolving)  arc  up  in  the  same  time  the*  the 
other  body  P  describes  the  similar  and  equal  arc  VP  in  the  quiescov.t  fig 
ure  YPK.  Find,  then,  by  Cor.  5,  Prop.  VI.,  the  centripetal  force  by  which 
the  body  may  be  made  to  revolve  in  the  curve  line  which  the  pom*  p  de 
scribes  in  an  immovable  plane,  and  the  Problem  will  be  solved.  O/E.K. 

PROPOSITION  XLIV.     THEOREM  XIV. 

The  difference  of  the  forces,  by  which  two  bodies  may  be  madi,  to  KMVG 
equally,  one  in  a  quiescent,  the  other  in  the  same  orbit  revolving,  i 1  in 
a  triplicate  ratio  of  their  common  altitudes  inversely. 
Let  the  parts  of  the  quiescent  or 
bit  VP,  PK  be  similar  and  equal  to 
the  parts  of  the  revolving  orbit  up, 
pk  ;  and  let  the  distance  of  the  points 
P  and  K  be  supposed  of  the  utmost 
smallness  Let  fall  a  perpendicular 
kr  from  the  point  k  to  the  right  line 
pC,  and  produce  it  to  m,  so  that  mr 
may  be  to  kr  as  the  angle  VC/?  to  the  /2\- 
angle  VCP.  Because  the  altitudes 
of  the  bodies  PC  and  pV,  KG  and 
kC}  are  always  equal,  it  is  manifest 
that  the  increments  or  decrements  of 
the  lines  PC  and  pC  are  always 
equal ;  and  therefore  if  each  of  the 
several  motions  of  the  bodies  in  the  places  P  and  p  be  resolved  into  two 
(by  Cor.  2  of  the  Laws  of  Motion),  one  of  which  is  directed  towards  the 
centre,  or  according  to  the  lines  PC,  pC,  and  the  other,  transverse  to  the 
former,  hath  a  direction  perpendicular  to  the  lines  PC  and  pC  ;  the  mo 
tions  towards  the  centre  will  be  equal,  and  the  transverse  motion  of  the 
body  p  will  be  to  the  transverse  motion  of  the  body  P  as  the  angular  mo 
tion  of  the  line  pC  to  the  angular  motion  of  the  line  PC ;  that  is,  as  the 
angle  VC/?  to  the  angle  VCP.  Therefore,  at  the  same  time  that  the  bodv 
P,  by  both  its  motions,  comes  to  the  point  K,  the  body  p,  having  an  equal 
motion  towards  the  centre,  will  be  equally  moved  from  p  towards  C  ;  arid 
therefore  that  time  being  expired,  it  will  be  found  somewhere  in  the 
line  mkr,  which,  passing  through  the  point  k,  is  perpendicular  to  the  line 
pC ;  and  by  its  transverse  motion  will  acquire  a  distance  from  the  line 


174  THE    MATHEMATICAL    PRINCIPLES  [BOOK  J. 

»C,  that  will  be  to  the  distance  which  the  other  body  P  acquires  from  the 
line  PC  as  the  transverse  motion  of  the  body  p  to  the  transverse  motion  of 
the  other  body  P.  Therefore  since  kr  is  equal  to  the  distance  which  the 
body  P  acquires  from  the  line  PC,  and  mr  is  to  kr  as  the  angle  VC/?  to 
the  angle  VCP,  that  is,  as  the  transverse  motion  of  the  body  p  to  the 
transverse  motion  of  the  body  P,  it  is  manifest  that  the  body  p,  at  the  ex 
piration  of  that  time,  will  be  found  in  the  place  m.  These  things  will  be 
so,  if  the  bodies  jo  and  P  are  equally  moved  in  the  directions  of  the  lines 
pC  and  PC,  and  are  therefore  urged  with  equal  forces  in  those  directions. 
I:  ut  if  we  take  an  angle  pCn  that  is  to  the  angle  pCk  as  the  angle  VGj0 
to  the  angle  VCP,  and  nC  be  equal  to  kG,  in  that  case  the  body  p  at  the 
expiration  of  the  time  will  really  be  in  n  ;  and  is  therefore  urged  with  a 
greater  force  than  the  body  P,  if  the  angle  nCp  is  greater  than  the  angle 
kCp,  that  is,  if  the  orbit  npk,  move  either  in  cmiseqnentia,  or  in  antece- 
denticij  with  a  celerity  greater  than  the  double  of  that  with  which  the  line 
CP  moves  in  conseqnentia  ;  and  with  a  less  force  if  the  orbit  moves  slower 
in  antecedent-la.  And  ihj  difference  of  the  forces  will  be  as  the  interval 
mn  of  the  places  through  which  the  body  would  be  carried  by  the  action  of 
that  difference  in  that  given  space  of  time.  About  the  centre  C  with  the 
interval  Cn  or  Ck  suppose  a  circle  described  cutting  the  lines  mr,  tun  pro 
duced  in  s  and  £,  and  the  rectangle  mn  X  nit  will  be  equal  to  the  rectan- 

*//?  n       ^*      */??  ^ 

•"•le  mk  X  ins,  and  therefore  mn  will  be  equal  to  —  — .       But   since 

mt 

the  triangles  pCk,  pCn,  in  a  given  time,  are  of  a  given  magnitude,  kr  and 
mr.  a  id  their  difference  mk,  and  their  sum  ms,  are  reciprocally  as  the  al 
titude  pC,  and  therefore  the  rectangle  mk  X  ms  is  reciprocally  as  the 
square  of  the  altitude  pC.  But,  moreover,  mt  is  directly  as  |//z/,  that  is,  as 
the  altitude  pC.  These  are  the  first  ratios  of  the  nascent  lines  ;  and  hence 

— r — -    that  is,  the  nascent  lineola  mn.  and  the  difference  of  the  forces 

mt 

proportional  thereto,  are  reciprocally  as  the  cube  of  the  altitude  pC. 
Q.E.D. 

COR.  I.  Hence  the  difference  of  the  forces  in  the  places  P  and  p,  or  K  and 
/.*,  is  to  the  force  with  which  a  body  may  revolve  with  a  circular  motion 
from  R  to  K,  in  the  same  time  that  the  body  P  in  an  immovable  orb  de 
scribes  the  arc  PK,  as  the  nascent  line  m,n  to  the  versed  sine  of  the  nascent 

mk  X  ms       rk2 

arc  RK,  that  is,  as — —  to  ^g,  or  as  mk  X  ms  to   the  square  of 

rk  ;  that  is.  if  we  take  given  quantities  F  and  G  in  the  same  ratio  to  one 
another  as  the  angle  VCP  bears  to  the  angle  VQ?,  as  GG  —  FF  to  FF. 
And,  therefore,  if  from  the  centre  C,  with  any  distance  CP  or  Cp,  there  be 
described  a  circular  sector  equal  to  the  whole  area  VPC,  which  the  body 


OEC. 


IX.l 


OF    NATURAL     PHILOSOPHY. 


175 


revolving  in  an  immovable  orbit  has  by  a  radius  drawn  to  the  centre  de- 
bribed  in  any  certain  time,  the  difference  of  the  forces,  with  which  the 
body  P  revolves  in  an  immovable  orbit,  and  the  body  p  in  a  movable  or 
bit,  will  be  to  the  centripetal  force,  with  which  another  body  by  a  radius 
drawn  to  the  centre  can  uniformly  describe  that  sector  in  the  same  time 
as  the  area  VPC  is  described,  as  GG  — FF  to  FF.  For  that  sector  and 
the  area  pCk  are  to  one  another  as  the  times  in  which  they  are  described. 

COR.  2.  If  the  orbit  YPK  be  an 
ellipsis,  having  its  focus  C,  and  its 
highest  apsis  Y,  and  we  suppose  the 
the  ellipsis  upk  similar  and  equal  to     .. 
it,  so  that  pC  may  be  always  equal  / 
to  PC,  and  the  angle  YC/?  be  to  the  •; 
angle  YCP  in  the  given  ratio  of  G   \ 
to  F  ;  and  for  the  altitude  PC  or  pC    \ 
we  put  A,  and  2R  for  the  latus  rec-   /t\ 
turn  of  the  ellipsis,  the  force  with       * 
which  a  body  may   be  made  to  re 
volve  in  a  movable  ellipsis  will  be  as 

FF       RGG  — RFF 

— -  +   -      — -rg —    — ,  and  vice  versa. 
/Y  A.  A. 

Let  the  force  with  which  a  body  may 

revolve  in  an  immovable  ellipsis  be  expressed  by  the  quantity       ,  and  the 


-.  7 


force  in  V  will  be 


FF 


But  the  force  with  which  a  body  may  revolve  in 


a  circle  at  the  distance  CY,  with  the  same  velocity  as  a  body  revolving  in 
an  ellipsis  has  in  Y,  is  to  the  force  with  which  a  body  revolving  in  an  ellip 
sis  is  acted  upon  in  the  apsis  Y,  as  half  the  latus  rectum  of  the  ellipsis  to  the 

RFF 

semi-diameter  CY  of  the  circle,  and  therefore  is  as    ,     =- :    and    tlu 


RFF 

which  is  to  this,  as  GG  —  FF  to  FF,  is  as  -    ~py^~~     ~:  and  this  force 

(by  Cor.  1  cf  this  Prop.)  is  the  difference  of  the  forces  in  Y,  with  which  the 
body  P  revolves  in  the  immovable  ellipsis  YPK,  and  the  body  p  in  the 
movable  ellipsis  upk.  Therefore  since  by  this  Prop,  that  difference  at 

any  other  altitude  A  is  to  itself  at  the  altitude  CY  as  -r-,  to  ^TF-»  the  same 

AJ      CYJ 

R  C^  ( "*  R  P^  T*1 

difference  in  every  altitude  A  will  be  as  -       —  3 —  : — .    Therefore  to  the 

FF 
force  -T-:  ,  by  which  the  body  may  revolve  in  an  immovable  ellipsis  VPK 


176  THE    MATHEMATICAL    PRINCIPLES  [BOOK    I. 

idd  the  excess  —    -:-= ,  and  the  sum  will  be  the  whole  force  -r-r  -\- 

A  AA 

RGG  —  RFF, 

.-5 —       by  which  a  body  may  revolve  in  the  same  time  in  the  mot- 

A. 

•able  ellipsis  upk. 

COR.  3.  In  the  same  manner  it  will  be  found,  that,  if  the  immovable  or 
bit  VPK  be  an  ellipsis  having  its  centre  in  the  centre  of  the  forces  C}  and 
there  be  supposed  a  movable  ellipsis  -upk,  similar,  equal,  and  concentrical 
to  it ;  and  2R  be  the  principal  latus  rectum  of  that  ellipsis,  and  2T  the 
latus  transversum,  or  greater  axis ;  and  the  angle  VCjo  be  continually  to  the 
angle  TCP  as  G  to  F ;  the  forces  with  which  bodies  may  revolve  in  the  im- 

FFA  FFA 

movable  and  movable  ellipsis,  in  equal  times,  will  be  as     ^      and  -p™~ 

RGG  — RFF 

+          —  .-3 respectively. 

A 

COR.  4.  And  universally,  if  the  greatest  altitude  CV  of  the  body  be  called 
T,  and  the  radius  of  the  curvature  which  the  orbit  VPK  has  in  Y,  that  is, 
the  radius  of  a  circle  equally  curve,  be  called  R,  and  the  centripetal  force 
with  which  a  body  may  revolve  in  any  immovable  trajectory  VPK  at  the  place 

VFF 
V  be    called    -f-=Trri ,  and  in  other  places  P  be  indefinitely  styled  X ;  and  the 

altitude  CP  be  called  A,  and  G  be  taken  to  F  in  the  given  ratio  of  the 
angle  VCjD  to  the  angle  VCP ;  the  centripetal  force  with  which  the  same 
body  will  perform  the  same  motions  in  the  same  time,  in  the  same  trajectory 
•upk  revolving  with  a  circular  motion,  will  be  as  the  sum  of  the  forces  X  -f- 
VRGG  — VRFF 

~  A* 

COR.  5.  Therefore  the  motion  of  a  body  in  an  immovable  orbit  being 
given,  its  angular  motion  round  the  centre  of  the  forces  may  be  increased 
or  diminished  in  a  given  ratio;  and  thence  new  immovable  orbits  may  be 
found  in  which  bodies  may  revolve  with  new  centripetal  forces. 

COR.  6.  Therefore  if  there  be  erected  the  line  VP  of  an  indeterminate 
-p  length,  perpendicular  to  the  line  CV  given  by  po 

sition,  and  CP  be  drawn,  and  Cp  equal  to  it,  mak 
ing  the  angle  VC/?  having  a  given  ratio  to  the  an 
gle  VCP,  the  force  with  which  a  body  may  revolve 
in  the  curve  line  Vjo/r,  which  the  point  p  is  con 
tinually  describing,  will  be  reciprocally  as  the  cube 
C  of  the  altitude  Cp.  For  the  body  P,  by  its  vis  in 
ertia  alone,  no  other  force  impelling  it,  will  proceed  uniformly  in  the  right 
line  VP.  Add,  then,  a  force  tending  to  the  centre  C  reciprocally  as  the 
cube  of  the  altitude  CP  or  Cp,  and  (by  what  was  just  demonstrated)  the 


SEC.    IX..J  OF    NATURAL    PHILOSOPHY.  177 

body  will  deflect  from  the  rectilinear  motion  into  the  curve  line  Ypk.  But 
this  curve  ~Vpk  is  the  same  with  the  curve  VPQ  found  in  Cor.  3,  Prop 
XLI,  in  which,  I  said,  hodies  attracted  with  such  forces  would  ascend 
obliquely. 

PROPOSITION  XLV.     PROBLEM  XXXL 

To  find  the  motion  of  the  apsides  in  orbits  approaching  very  near  to 

circles. 

This  problem  is  solved  arithmetically  by  reducing  the  orbit,  which  a 
body  revolving  in  a  movable  ellipsis  (as  in  Cor.  2  and  3  of  the  above 
Prop.)  describes  in  an  immovable  plane,  to  the  figure  of  the  orbit  whose 
apsides  are  required ;  and  then  seeking  the  apsides  of  the  orbit  which  that 
body  describes  in  an  immovable  plane.  But  orbits  acquire  the  same  figure, 
if  the  centripetal  forces  with  which  they  are  described,  compared  between 
themselves,  are  made  proportional  at  equal  altitudes.  Let  the  point  V  be 
the  highest  apsis,  and  write  T  for  the  greatest  altitude  CV,  A  for  any  other 
altitude  CP  or  C/?,  and  X  for  the  difference  of  the  altitudes  C V  —  CP : 
and  the  force  writh  which  a  body  moves  in  an  ellipsis  revolving  about  its 

•p  -p       T?  C*  f  ^ T?  F*  F 

focus  C  (as  in  Cor.  2),  and  which  in  Cor.  2  was  as  -r-r  -\ —   — -.-3 —  — , 

FFA  +  RGG  —  RFF  , 

that  is  as, -^ — ,  by  substituting  T  —  X  for  A,  will  be- 

A 

RGG  —  RFF  +  TFF  —  FFX 

come  as -p — .  In  like  manner  any  other  cen 
tripetal  force  is  to  be  reduced  to  a  fraction  whose  denominator  is  A3,  and 
the  numerators  are  to  be  made  analogous  by  collating  together  the  homo 
logous  terms.  This  will  be  made  plainer  by  Examples. 

EXAMPLE  1.    Let   us   suppose   the   centripetal    force   to   be   uniform, 

A3 
and  therefore   as  —3  or,   writing    T  —  X  for  A  in    the    numerator,  as 

T3  —  3TTX  +  3TXX  —  X3 

—  —         — =-.      Ihen  collating  together  the  correspon- 

A3 

dent  terms  of  the  numerators,  that  is,  those  that  consist  of  given  quantities, 
with  those  of  given  quantities,  and  'those  of  quantities  not  given  with  those 
of  quantities  not  given,  it  will  become  RGG  —  RFF  -f-  TFF  to  T3  as  — 
FFX  to  3TTX  -f  3TXX  —  X3,  or  as  — FF  to  —  3TT  +  3TX  —  XX. 
Now  since  the  orbit  is  supposed  extremely  near  to  a  circle,  let  it  coincide 
with  a  circle ;  and  because  in  that  case  R  and  T  become  equal,  and  X  is 
infinitely  diminished,  the  last  ratios  will  be,  as  RGG  to  T2,  so  —  FF  to  — 
3TT,  or  as  GG  to  TT,  so  FF  to  3TT;  and  again,  as  GG  to  FF,  so  TT 
to  3TT,  that  is,  as  1  to  3  ;  and  therefore  G  is  to  F,  that  is,  the  angle  VC/? 
to  the  angle  VCP,  as  1  to  v/3.  Therefore  since  the  body,  in  an  immovable 


178  THE    MATHEMATICAL    PRINCIPLES  [BOOK    I 

ellipsis,  in  descending  from  the  upper  to  the  lower  apsis,  describes  an  angle, 
if  I  may  so  speak,  of  ISO  deg.,  the  other  body  in  a  movable  ellipsis,  and  there 
fore  in  the  immovable  orbit  we  are  treating  of,  will  in  its  descent  from 

180 

the  upper  to  the  lower  apsis,  describe  an  angle  VCjt?  of  — ^  deg.     And  this 

\/o 

comes  to  pass  by  reason  of  the  likeness  of  this  orbit  which  a  body  acted 
upon  by  an  uniform  centripetal  force  describes,  and  of  that  orbit  which  a 
body  performing  its  circuits  in  a  revolving  ellipsis  will  describe  in  a  quies 
cent  plane.  By  this  collation  of  the  terms,  these  orbits  are  made  similar ; 
not  universally,  indeed,  but  then  only  when  they  approach  very  near  to  a 
circular  figure.  A  body,  therefore  revolving  with  an  uniform  centripetal 

180 

force  in  an  orbit  nearly  circular,  will  always  describe  an  angle  of  — »  deg/,  or 

v/o 

103  deg.,  55  m.,  23  sec.,  at  the  centre;  moving  from  the  upper  apsis  to  the 
lower  apsis  when  it  has  once  described  that  angle,  and  thence  returning  to 
the  upper  apsis  when  it  has  described  that  angle  again ;  and  so  on  in  in- 
finitwn. 

EXAM.  2.  Suppose  the  centripetal  force  to  be  as  any  power  of  the  alti- 

An 

tude  A,  as,  for  example,  An — 3,  or-r^  ;  where  n  —  3  and  n  signify  any  in- 

A. 

dices  of  powers  whatever,  whether  integers  or  fractions,  rational  or  surd, 
affirmative  or  negative.  That  numerator  An  or  T  —  X|n  being  reduced  to 
an  indeterminate  series  by  my  method  of  converging  series,  will  become 

Tn  —  >/XTn— T  +  •  —  ^  XXTn— 2,  &c.  And  conferring  these  terms 
with  the  terms  of  the  other  numerator  RGG  — RFF  +  TFF  —  FFX,  it 
becomes  as  RGG  —  RFF  4-  TFF  to  Tn,  so  —  FF  to  —  ?/.Tn— r  +  ?~^ 

XTn — 2,  &c.  And  taking  the  last  ratios  where  the  orbits  approach  to 
circles,  it  becomes  as  RGG  to  T'1,  so  —  FF  to  —  nT-1— T,  or  as  GG  to 
T"— ',  so  FF  to  ?*Tn—  ;  and  again,  GG  to  FF,  so  Tn— l  to  nT"—1,  that 
is,  as  1  to  n  ;  and  therefore  G  is  to  F,  that  is  the  angle  VCp  to  the  angle 
VCP,  as  1  to  ^/n.  Therefore  since  the  angle  VCP,  described  in  the  de 
scent  of  the  body  from  the  upper  apsis  to  the  lower  apsis  in  an  ellipsis,  is 
of  180  deg.,  the  angle  VC/?,  described  in  the  descent  of  the  body  from  the 
upper  apsis  to  the  lower  apsis  in  an  orbit  nearly  circular  which  a  body  de 
scribes  with  a  centripetal  force  proportional  to  the  power  An — 3,  will  be  equal 

ISO 

to   an  angle  of  -  —  deg.,  and  this  angle  being  repeated,  the  body  will  re- 
\/ti 

turn  from  the  lower  to  the  upper  apsis,  and  so  on  in  infinitum.  As  if  the 
centripetal  force  be  as  the  distance  of  the  body  from  the  centre,  that  is,  as  A, 

A4 
or  -p,  n  will  be  equal  to  4,  and  ^/n  equal  to  2  ;  and  thereLre  the  angle 


IX.]  OF    NATURAL    PHILOSOPHY.  IT9 

ISO 

between  the  upper  and  the  lower  apsis  will  be  equal  to  —  deg.,  or  90  deg. 

Therefore  the  body  having  performed  a  fourth  part  of  one  revolution,  will 
arrive  at  the  lower  apsis,  and  having  performed  another  fourth  part,  will 
arrive  at  the  upper  apsis,  and  so  on  by  turns  in  infiuitum.  This  appears 
also  from  Prop.  X.  For  a  body  acted  on  by  this  centripetal  force  will  re 
volve  in  an  immovable  ellipsis,  whose  centre  is  the  centre  of  force.  If  the 

1       A2 

centripetal  force  is  reciprocally  as  the  distance,  that  is,  directly  as  —  or  — 

£     '  A       A"' 

ji  will  be  equal  to  2 ;  and  therefore  the  angle  between  the  upper  and  lower 

180 

apsis  will  be  — -  deg.,  or  127  deg.,  16  min.,  45  sec. ;  and  therefore  a  body  re 
v/2 

volving  with  such  a  force,  will  by  a  perpetual  repetition  of  this  angle,  move 
alternately  from  the  upper  to  the  lower  and  from  the  lower  to  the  upper 
apsis  for  ever.  So.  also,  if  the  centripetal  force  be  reciprocally  as  the 
biquadrate  root  of  the  eleventh  power  of  the  altitude,  that  is,  reciprocally 

as  A  —,  and,  therefore,  directly  as  -r-fp  or  as  Ts>  n  wil*  ^e  etlual   f°  \>  an(1 
4  A^-  A 

1  Of) 

-  deg.  will  be  equal  to  360  deg. ;  and  therefore  the  body  parting  from 
v/ n 

the  upper  apsis,  and  from  thence  perpetually  descending,  will  arrive  at  the 
lower  apsis  when  it  has  completed  one  entire  revolution ;  and  thence  as 
cending  perpetually,  when  it  has  completed  another  entire  revolution,  it 
will  arrive  again  at  the  upper  apsis ;  and  so  alternately  for  ever. 

EXAM.  3.    Taking  m  and  n  for  any  indices  of  the  powers  of  the  alti 
tude,  and  b  and  c  for  any  given  numbers,  suppose  the  centripetal   force 

6Ara  +  cA"  b   into    T  —  X>  -f-  c  into    T  —  X 

to  be  as r^ that   is,  as 


A3  A3 

or     (by     the     method     of     converging      series      above-mentioned)     as 

bTm  +  cTn  —  m6XT"  -1     //cXTn—  '      mm  —  m   vvrpm  un  —  n 

~~2  --  0A.A1  —  ^— 

— 

t-XXT"  —  2,  «fcc. 

—  T$~—      ~  and  comparing   the  terms  of  the  numerators,  there  will 

arise  RGG  —  II  FF  -f  TFF  to  ^Tm  +  cT"  as  —  FF  to  —  mbTm  —  i  — 


"  -  «  +          2"  m  bXT"  -  *  +  "^p  cXTn  -  »  .fee.     And  tak- 

ing  the  last  ratios  that  arise  when  the  orbits  come  to  a  circular  form,  there 
will  come  forth  GG  to  6Tm  —  l  -f  cTn  —  1  as  FF  to  mbTm  —  l  +  ncT"  —  J  ; 
and  again,  GG  to  FF  as  6Tm  —  '  +  cTn  —  '  to  mbTn  —  1  -f  ncTn  —  \ 
This  proportion,  by  expressing  the  greatest  altitude  CV  or  T  arithmeti 
cally  by  unity,  becomes,  GG  to  FF  as  b  -{-  c  to  mb  -\-  ?/c,  and  therefore  as  I 


(80  THE    MATHEMATICAL    PRINCIPLES  [BOOK    1 

tub  ~h  nc 
to  -  y—7— — •     Whence  G  becomes  to  P,  that  is,  the  angle  VCjo  to  the  an- 

f)   ~T~   C 

gle  VCP.  as  1  to  >/-  .       -'-.     And  therefore  since  the  angle  VCP  between 

the  upper  and  the  lower  apsis,  in  an  immovable  ellipsis,  is  of  180  deg.,  thr 
angle  VC/?  between  the  same  apsides  in  an  orbit  which  a  body  describes 

b  A  m    I    c  A  n 

with  a  centripetal  force,  that  is.  as  -  — r§— ,  will  be  equal  to  an  angle  of 

A. 

ISO  v/— 1~TT~ ;  deg.  And  °y  tne  same  reasoning,  if  the  centripetal  force 
be  as  -  —73 ,  the  angle  between  the  apsides  will  be  found  equal  to 

fi  f* 

18o  V'' — - —      -  deg.     After  the  same  manner  the  Problem  is  solved  in 
nib  —  >ic 

more  difficult  cases.  The  quantity  to  which  the  centripetal  force  is  pro 
portional  must  always  be  resolved  into  a  converging  series  whose  denomi 
nator  is  A*.  Then  the  given  part  of  the  numerator  arising  from  that 
operation  is  to  be  supposed  in  the  same  ratio  to  that  part  of  it  which  is  not 
given,  as  the  given  part  of  this  numerator  RGG  —  RFF  -f  TFF  —  FFX. 
is  to  that  part  of  the  same  numerator  which  is  not  given.  And  taking 
away  the  superfluous  quantities,  and  writing  unity  for  T,  the  proportion 
of  G  to  F  is  obtained. 

COR.  1 .  Hence  if  the  centripetal  force  be  as  any  power  of  the  altitude, 
that  power  may  be  found  from  the  motion  of  the  apsides ;  and  so  contra 
riwise.  That  is,  if  the  whole  angular  motion,  with  which  the  body  returns 
to  the  same  apsis,  be  to  the  angular  motion  of  one  revolution,  or  360  deg., 
MS  any  number  as  m  to  another  as  n,  and  the  altitude  called  A ;  the  force 

nn 

will  be  as  the  power  A  HSii  — 3  of  the  altitude  A;  the  index  of  which  power  is 
-  —  3.     This  appears  by  the  second  example.     Hence  it  is  plain  that 

the  force  in  its  recess  from  the  centre  cannot  decrease  in  a  greater  than  a 
triplicate  ratio  of  the  altitude.  A  body  revolving  with  such  a  force,  and 
parting  from  the  apsis,  if  it  once  begins  to  descend,  can  never  arrive  at  the 
lower  apsis  or  least  altitude,  but  will  descend  to  the  centre,  describing  the 
curve  line  treated  of  in  Cor.  3,  Prop.  XLL  But  if  it  should,  at  its  part 
ing  from  the  lower  apsis,  begin  to  ascend  never  so  little,  it  will  ascend  in 
irtfimtifm,  and  never  come  to  the  upper  apsis ;  but  will  describe  the  curve 
line  spoken  of  in  the  same  Cor.,  and  Cor.  6,  Prop.  XLIV.  So  that  where 
the  force  in  its  recess  from  the  centre  decreases  in  a  greater  than  a  tripli 
cate  ratio  of  the  altitude,  the  body  at  its  parting  from  the  apsis,  will  either 
descend  to  the  centre,  or  ascend  in  iiiftnitum,  according  as  it  descends  or 
Ascends  at  the  beginning  of  its  motion.  But  if  the  force  in  its  recess  from 


"SEC.    IX.J  OF    NATURAL    PHILOSOPHY.  ISi 

the  centre  either  decreases  in  a  less  than  a  triplicate  ratio  of  the  altitude, 
or  increases  in  any  ratio  of  the  altitude  whatsoever,  the  body  will  never 
descend  to  the  centre,  but  will  at  some  time  arrive  at  the  lower  apsis  ;  and, 
on  the  contrary,  if  the  body  alternately  ascending  and  descending  from  one 
apsis  to  another  never  comes  to  the  centre,  then  either  the  force  increases 
in  the  recess  from  the  centre,  or  it  decreases  in  a  less  than  a  triplicate  ratio 
of  the  altitude;  and  the  sooner  the  body  returns  from  one  apsis  to  another, 
the  farther  is  the  ratio  of  the  forces  from  the  triplicate  ratio.  As  if  the 
body  should  return  to  and  from  the  upper  apsis  by  an  alternate  descent  and 
ascent  in  8  revolutions,  or  in  4,  or  2,  or  \\  •  that  is,  if  m  should  be  to  n  as  8, 

or  4,  or  2,  or  H  to  1.  and  therefore  ---  3,  be  g\  —  3,or  TV~  3,  or  i  —  3,  or 

mm 

3 


I  -  3  ;  then  the  force  will  be  as  A~  ?  or  AT°  "~  3j  or  A*~~  3j  or  A""  G  ' 

that  is.  it  will  be  reciprocally  as  A3  C4'  or  A3  T^'  or  A3  4'  or  A3  "" 
If  the  body  after  each  revolution  returns  to  the  same  apsis,  and  the  apsis 

nn    _ 

remains  unmoved,  then  m  will  be  to  n  as  1  to  1,  and  therefore  A»"« 

will  be  equal  to  A     2,  or  —  -  ;  and  therefore  the  decrease  of  the  forces  will 

A  A 

be  in  a  duplicate  ratio  of  the  altitude  ;  as  was  demonstrated  above.  If  the 
body  in  three  fourth  parts,  or  two  thirds,  or  one  third,  or  one  fourth  part 
of  an  entire  revolution,  return  to  the  same  apsis  ;  m  will  be  to  n  as  £  or  ? 

n  n  i_6  _  3  9  _  3  o 

or  ^  or  i  to  1,  and  therefore  Amm     3  is  equal  to  A  9        'or  A4         or  A 

_  3  1  6  _  3  l_l 

'  '  or  A  ;  and  therefore  the  force  is  either  reciprocally  as  A  fl      or 

3  613 

A4'  or  directly  as  A  or  A  .  Lastly  if  the  body  in  its  progress  from  the 
upper  apsis  to  the  same  upper  apsis  again,  goes  over  one  entire  revolution 
and  three  deg.  more,  and  therefore  that  apsis  in  each  revolution  of  the  body 
moves  three  deg.  in  consequentia  ;  then  m  will  be  to  u  as  363  deg.  to 


360  deg.  or  as   121   to   120,  and  therefore   Amm  will   be    equal    to 

2  9_  5_  2_  JJ 

A  "'  and   therefore  the  centripetal  force   will    be   reciprocally  as 

^T4"6TT>  or  reciprocally  as  A2^4^  very  nearly.  Therefore  the  centripetal 
force  decreases  in  a  ratio  something  greater  than  the  duplicate ;  but  ap 
proaching  59f  times  nearer  to  the  duplicate  than  the  triplicate. 

COR.  2.  Hence  also  if  a  body,  urged  by  a  centripetal  force  which  is  re 
ciprocally  as  the  square  of  the  altitude,  revolves  in  an  ellipsis  whose  focus 
is  in  the  centre  of  the  forces ;  and  a  new  and  foreign  force  should  be  added 
to  or  subducted  from  this  centripetal  force,  the  motion  of  the  apsides  arising 
from  that  foreign  force  may  (by  the  third  Example)  be  known ;  and  so  on 
the  contrary.  As  if  the  force  with  which  the  body  revolves  in  the  ellipsis 


182  THE    MATHEMATICAL    PRINCIPLES  [BOOK    I 

oe  as  -r-r- ;  and  the  foreign  force  subducted  as  cA,  and  therefore  the  remain- 

A  .A. 

^ c^4 

ing  force  as  — -^ ;  then  (by  the  third  Example)  b  will  be  equal  to   1. 

m  equal  to  1,  and  n  equal  to  4 ;  and  therefore  the  angle  of  revolution  be 

1  —  c 

•tween  the  apsides  is  equal  to  180  <*/- —  deg.     Suppose  that  foreign  force 

to  be  357.45  parts  less  than  the  other  force  with  which  the  body  revolves 
in  the  ellipsis  :  that  is,  c  to  be  -3  }y£  j  ;  A  or  T  being  equal  to  1  ;  and  then 

l8(Vl~4c  will  be  18<Vfff  Jf  or  180.7623,  that  is,  180  deg.,  45  min., 

44  sec.  Therefore  the  body,  parting  from  the  upper  apsis,  will  arrive  at 
the  lower  apsis  with  an  angular  motion  of  180  deg.,  45  min.,  44  sec  ,  and 
this  angular  motion  being  repeated,  will  return  to  the  upper  apsis ;  and 
therefore  the  upper  apsis  in  each  revolution  will  go  forward  1  deg.,  31  min., 
28  sec.  The  apsis  of  the  moon  is  about  twice  as  swift 

So  much  for  the  motion  of  bodies  in  orbits  whose  planes  pass  through 
the  centre  of  force.  It  now  remains  to  determine  those  motions  in  eccen 
trical  planes.  For  those  authors  who  treat  of  the  motion  of  heavy  bodies 
used  to  consider  the  ascent  and  descent  of  such  bodies,  not  only  in  a  per 
pendicular  direction,  but  at  all  degrees  of  obliquity  upon  any  given  planes  ; 
and  for  the  same  reason  we  are  to  consider  in  this  place  the  motions  of 
bodies  tending  to  centres  by  means  of  any  forces  whatsoever,  when  those 
bodies  move  in  eccentrical  planes.  These  planes  are  supposed  to  be 
perfectly  smooth  and  polished,  so  as  not  to  retard  the  motion  of  the  bodies 
in  the  least.  Moreover,  in  these  demonstrations,  instead  of  the  planes  upon 
which  those  bodies  roll  or  slide,  and  which  are  therefore  tangent  planes  to 
the  bodies,  I  shall  use  planes  parallel  to  them,  in  which  the  centres  of  the 
bodies  move,  and  by  that  motion  describe  orbits.  And  by  the  same  method 
I  afterwards  determine  the  motions  of  bodies  performer1  in  curve  superficies. 


SECTION  X. 

Of  the  motion  of  bodies  in  given  superficies,  and  of  the  reciprocal  motion 
of fnnependulous  bodies. 

PROPOSITION  XL VI.     PROBLEM  XXXII. 

Any  kind  of  centripetal  force  being  supposed,  and  the  centre  of 'force •,  atfft 
any  plane  whatsoever  in  which  the  body  revolves,  being  given,  and  tint 
quadratures  of  curvilinear  figures  being  allowed;  it  is  required  to  de 
termine  the  motion  of  a  body  going  off  from  a  given  place.,  with  a 
given  velocity,  in  the  direction  of  a  given  right  line  in,  that  plane. 


SEC.  X.J  OF    NATURAL    PHILOSOPHY-  183 

Let  S  be  the  centre  of  force,  SC  the 
least  distance  of  that  centre  from  the  given 
plane,  P  a  body  issuing  from  the  place  P 
in  the  direction  of  the  right  line  PZ,  Q, 
the  same  body  revolving  in  its  trajectory, 
and  PQ,R  the  trajectory  itself  which  is 
required  to  be  found,  described  in  that 
given  plane.  Join  CQ,  Q.S,  and  if  in  Q,S 
we  take  SV  proportional  to  the  centripetal 
force  with  which  the  body  is  attracted  to 
wards  the  centre  S,  and  draw  VT  parallel 
to  CQ,  and  meeting  SC  in  T  ;  then  will  the  force  SV  be  resolved  into 
two  (by  Cor.  2,  of  the  Laws  of  Motion),  the  force  ST,  and  the  force  TV  ;  of 
which  ST  aMracting  the  body  in  the  direction  of  a  line  perpendicular  to 
that  plane,  does  not  at  all  change  its  motion  in  that  plane.  But  the  action 
c  f  the  other  force  TV,  coinciding  with  the  position  of  the  plane  itself,  at 
tracts  the  body  directly  towards  the  given  point  C  in  that  plane ;  a«d 
t  icreftre  causes  the  body  to  move  in  this  plane  in  the  same  manner  as  if 
the  force  S  F  were  taken  away,  and  the  body  were  to  revolve  in  free  space 
about  the  centre  C  by  means  of  the  force  TV  alone.  But  there  being  given 
the  centripetal  force  TV  with  which  the  body  Q,  revolves  in  free  space 
about  the  given  centre  C,  there  is  given  (by  Prop.  XLII)  the  trajectory 
PQ.R  which  the  body  describes ;  the  place  Q,,  in  which  the  body  will  be 
found  at  any  given  time ;  and,  lastly,  the  velocity  of  the  body  in  that  place 
Q,.  And  so  e  contra.  Q..E.I. 

PROPOSITION  XLV1L     THEOREM  XV. 

Supposing  the  centripetal  force  to  be  proportional  to  t/ie  distance  of  the 
body  from  the  centre  ;  all  bodies  revolving  in  any  planes  whatsoever 
will  describe  ellipses,  and  complete  their  revolutions  in  equal  times ; 
and  those  which  move  in  right  lines,  running  backwards  and  forwards 
alternately ',  will  complete  ttieir  several  periods  of  going  and  returning 
in  the  same  times. 

For  letting  all  things  stand  as  in  the  foregoing  Proposition,  the  force 
SV,  with  which  the  body  Q,  revolving  in  any  plane  PQ,R  is  attracted  to 
wards  the  centre  S,  is  as  the  distance  SO.  ;  and  therefore  because  SV  and 
SQ,,  TV  and  CQ,  are  proportional,  the  force  TV  with  which  the  body  is 
attracted  towards  the  given  point  C  in  the  plane  of  the  orbit  is  as  the  dis 
tance  CQ,.  Therefore  the  forces  with  which  bodies  found  in  the  plane 
PQ,R  are  attracted  towaitis  the  point  O,  are  in  proportion  to  the  distances 
equal  to  the  forces  with  which  the  same  bodies  are  attract-ed  every  way  to 
wards  the  centre  S  ;  and  therefore  the  bodies  will  move  in  the  same  times, 
and  in  the  same  figures,  in  any  plane  PQR  about  the  point  C.  n*  they 


THE    MATHEMATICAL    PRINCIPLES  [BOOK   I. 

would  do  in  free  spaces  about  the  centre  S ;  and  therefore  (by  Cor.  2,  Prop. 
X,  ai  d  Cor.  2,  Prop.  XXXVIII.)  they  will  in  equal  times  either  describe 
ellipses  m  that  plane  about  the  centre  C,  or  move  to  and  fro  in  right  lines 
passing  through  the  centre  C  in  that  plane;  completing  the  same  periods 
of  time  in  all  cases.  Q.E.D. 

SCHOLIUM. 

The  ascent  and  descent  of  bodies  in  curve  superficies  has  a  near  relation 
to  these  motions  we  have  been  speaking  of.  Imagine  curve  lines  to  be  de 
scribed  on  any  plane,  and  to  revolve  about  any  given  axes  passing  through 
the  centre  of  force,  and  by  that  revolution  to  describe  curve  superficies  ;  and 
that  the  bodies  move  in  such  sort  that  their  centres  may  be  always  found 
m  those  superficies.  If  those  bodies  reciprocate  to  and  fro  with  an  oblique 
ascent  and  descent,  their  motions  will  be  performed  in  planes  passing  through 
tiie  axis,  and  therefore  in  the  curve  lines,  by  whose  revolution  those  curve 
superficies  were  generated.  In  those  cases,  therefore,  it  will  be  sufficient  to 
consider  thp  motion  in  those  curve  lines. 

PROPOSITION  XL VIII.     THEOREM  XVI. 

If  a  wheel  stands  npon  the  outside  of  a  globe  at  right  angles  thereto,  and 
revolving  about  its  own  axis  goes  forward  in  a  great  circle,  the  length 
of  lite  curvilinear  path  which  any  point,  given  in  the  perimeter  of  the 
wheel,  hath  described,  since,  the  time  that  it  touched  the  globe  (which 
curvilinear  path  w~e  may  call  the  cycloid,  or  epicycloid),  will  be  to  double 
the  versed  sine  of  half  the  arc  which  since  that  time  has  touched  the 
globe  in  passing  over  it,  as  the  sn,m  of  the  diameters  of  the  globe  and 
the  wheel  to  the  semi-diameter  of  the  globe. 

PROPOSITION  XLIX.     THEOREM  XVII. 

ff  a  wheel  stand  upon  the  inside  of  a  concave  globe  at  right  angles  there 
to,  and  revolving  about  its  own  axis  go  forward  in  one  of  the  great 
circles  of  the  globe,  the  length  of  the  curvilinear  path  which  any  point, 
given  in  the  perimeter  of  the  wheel^  hath  described  since  it  toncJied  the 
globe,  imll  be  to  the  double  of  the  versed  sine  of  half  the  arc  which  in 
all  that  time  has  touched  the  globe  in  passing  over  it,  as  the  difference 
of  the  diameters  of  the  globe  and  the  wheel  to  the  semi-diameter  of  the 
globe. 

Let  ABL  be  the  globe.  C  its  centre,  BPV  the  wheel  insisting  thereon, 
E  the  centre  of  the  wheel,  B  the  point  of  contact,  and  P  the  given  point 
in  the  perimeter  of  the  wheel.  Imagine  this  wheel  to  proceed  in  the  great 
circle  ABL  from  A  through  B  towards  L,  and  in  its  progress  to  revolve  in 
such  a  manner  that  the  arcs  AB,  PB  may  be  always  equal  one  to  the  other, 
:if;d  the  given  point  P  in  the  peri  meter  of  the  wheel  may  describe  in  thf 


SEC.  X.I 


OF    NATURAL    PHILOSOPHY. 

s 


185 


H 


mean  time  the  curvilinear  path  AP.  Let  AP  be  the  whole  curvilinear 
path  described  since  the  wheel  touched  the  globe  in  A,  and  the  length  cf 
this  path  AP  will  be  to  twice  the  versed  sine  of  the  arc  |PB  as  20 E  to 
CB.  For  let  the  right  line  CE  (produced  if  need  be)  meet  the  wheel  in  V, 
and  join  CP,  BP,  EP,  VP  ;  produce  CP,  and  let  fall  thereon  the  perpen 
dicular  VF.  Let  PH,  VH,  meeting  in  H,  touch  the  circle  in  P  and  V, 
and  let  PH  cut  YF  in  G,  and  to  VP  let  fall  the  perpendiculars  GI,  HK. 
From  the  centre  C  with  any  interval  let  there  be  described  the  circle  wow, 
cutting  the  right  line  CP  in  nt  the  perimeter  of  the  wheel  BP  in  o,  and 
the  curvilinear  path  AP  in  m  ;  and  from  the  centre  V  with  the  interval 
Vo  let  there  be  described  a  circle  cutting  VP  produced  in  q. 

Because  the  wheel  in  its  progress  always  revolves  about  the  point  of  con 
tact  B.  it  is  manifest  that  the  right  line  BP  is  perpendicular  to  that  curve  line 
AP  which  the  point  P  of  the  wheel  describes,  and  therefore  that  the  right 
line  VP  will  touch  this  curve  in  the  point  P.  Let  the  radius  of  the  circle  nmn 
be  gradually  increased  or  diminished  so  that  at  last  it  become  equal  to  the 
distance  CP ;  and  by  reason  of  the  similitude  of  the  evanescent  figure 
Pnn-mq,  and  the  figure  PFGVI,  the  ultimate  ratio  of  the  evanescent  lined  ae 
Pra,  P//,  Po,  P<y,  that  is,  the  ratio  of  the  momentary  mutations  of  the  curve 
AP,  the  right  line  CP,  the  circular  arc  BP,  and  the  right  line  VP,  will  ••< 


iSS  THE    MATHEMATICAL    PRINCIPLES  [BOOK    1. 

the  same  as  of  the  lines  PV,  PF,  PG,  PI,  respectively.  But  since  VF  is 
perpendicular  to  OF,  and  VH  to  CV,  and  therefore  the  angles  HVG,  VCF 
equal:  and  the  angle  VHG  (because  the  angles  of  the  quadrilateral  figure 
HVEP  are  right  in  V  and  P)  is  equal  to  the  angle  CEP,  the  triangles 
V  HG,  CEP  will  be  similar ;  and  thence  it  will  come  to  pass  that  as  EP  is 
to  CE  so  is  HG  to  HV  or  HP,  and  so  KI  to  KP,  and  by  composition  or 
division  as  CB  to  CE  so  is  PI  to  PK,  and  doubling  the  consequents  asCB 
to  2CE  so  PI  to  PV,  and  so  is  Pq  to  Pm.  Therefore  the  decrement  of  the 
line  VP,  that  is,  the  increment  of  the  line  BY— VP  to  the  increment  of  the 
curve  line  AP  is  in  a  given  ratio  of  CB  to  2CE,  and  therefore  (by  Cor. 
Lena.  IV)  the  lengths  BY— YP  and  AP,  generated  by  those  increments,  arc 
in  the  same  ratio.  But  if  BY  be  radius,  YP  is  the  cosine  of  the  angle  BYP 
or  -*BEP,  and  therefore  BY— YP  is  the  versed  sine  of  the  same  angle,  and 
therefore  in  this  wheel,  whose  radius  is  ^BV,  BY— YP  will  be  double  the 
versed  sine  of  the  arc  ^BP.  Therefore  AP  is  to  double  the  versed  sine  oi 
the  arc  ^BP  as  2CE  to  CB.  Q.E.D. 

The  line  AP  in  the  former  of  these  Propositions  we  shall  name  the  cy 
cloid  without  the  globe,  the  other  in  the  latter  Proposition  the  cycloid  within 
the  globe,  for  distinction  sake. 

COR.  1.  Hence  if  there  be  described  the  entire  cycloid  ASL,  and  the 
same  be  bisected  in  S,  the  lencth  of  the  part  PS  will  be  to  the  length  PV 
(which  is  the  double  of  the  sine  of  the  angle  YBP,  when  EB  is  radius)  as 
2CE  to  CB,  and  therefore  in  a  given  ratio. 

COR.  2.  And  the  length  of  the  semi-perimeter  of  the  cycloid  AS  will  be 
equal  to  a  right  line  which  is  to  the  dumeter  of  the  wheel  BY  as  2CF 
toCB. 

PROPOSITION  L.     PROBLEM  XXXIII. 

To  cause  a  pendulous  body  to  oscillate  in  a  given  cycloid. 

Let  there  be  given  within  the  globe  QYS  de- 
scribed  with  the  centre  C,  the  cycloid  QRS,  bi 
sected  in  R,  and  meeting  the  superficies  of  the 
globe  with  its  extreme  points  Q  and  S  on  either 
hand.  Let  there  be  drawn  CR  birxcting  the  arc 
QS  in  O,  and  let  it  be  produced  to  A  in  such 
sort  that  CA  may  be  to  CO  as  CO  to  CR. 
About  the  centre  C,  with  the  interval  CA,  let 
there  be  described  an  exterior  globe  DAF  ;  and 
within  this  globe,  by  a  wheel  whose  diameter  is 
AO,  let  there  be  described  two  semi-cycloids  AQ,, 
AS,  touching  the  interior  globe  in  Q,  and  S,  and  meeting  the  exterior  globe 
in  A.  From  that  point  A,  with  a  thread  APT  in  length  equal  to  the  line 
AR,  let  the  body  T  depend,  and  oscillate  in  such  manner  between  the  two 


SlCC.    X.J  OF    NATURAL    PHILOSOPHY.  187 

semi-cycloids  AQ,  AS,  that,  as  often  as  the  pendulum  parts  from  the  per 
pendicular  AR,  the  upper  part  of  the  thread  AP  may  be  applied  to  that 
semi-cycloid  APS  towards  which  the  motion  tends,  and  fold  itself  round 
that  curve  line,  as  if  it  were  some  solid  obstacle,  the  remaining  part  of  the 
same  thread  PT  which  has  not  yet  touched  the  semi-cycloid  continuing 
straight.  Then  will  the  weight  T  oscillate  in  the  given  cycloid  QRS. 

Q.E.F. 

For  let  the  thread  PT  meet  the  cycloid  QRS  in  T,  and  the  circle  QOS 
m  V,  and  let  0V  be  drawn  j  and  to  the  rectilinear  part  of  the  thread  PT 
from  the  extreme  points  P  and  T  let  there  be  erected  the  perpendiculars 
BP,  T  W,  meeting  the  right  line  CV  in  B  and  W.  It  is  evident,  from  the 
construction  and  generation  of  the  similar  figures  AS,  SR,  that  those  per 
pendiculars  PB,  TVV,  cut  off  from  CV  the  lengths  VB,  VVV  equal  the 
diameters  of  the  wheels  OA,  OR.  Therefore  TP  is  to  VP  (which  is  dou 
ble  the  sine  of  the  angle  VBP  when  ^BV  is  radius)  as  BYV  to  BV,  or  AO 
-f-OR  to  AO,  that  is  (since  CA  and  CO,  CO  and  CR;  and  by  division  AO 
and  OR  are  proportional),  as  CA  +  CO  to  CA,  or,  if  BV  be  bisected  in  E, 
as  2CE  to  CB.  Therefore  (by  Cor.  1,  Prop.  XLIX),  the  length  of  the 
rectilinear  part  of  the  thread  PT  is  always  equal  to  the  arc  of  the  cycloid 
PS,  and  the  whole  thread  APT  is  always  equal  to  the  half  of  the  cycloid 
APS,  that  is  (by  Cor.  2,  Prop.  XLIX),  to  the  length  AR.  And  there 
fore  contrariwise,  if  the  string  remain  always  equal  to  the  length  AR,  the 
point  T  will  always  move  in  the  given  cycloid  QRS.  Q.E.D. 

COR.  The  string  AR  is  equal  to  the  semi-cycloid  AS,  and  therefore  has 
the  same  ratio  to  AC  the  semi-diameter  of  the  exterior  globe  as  the  like 
semi-cycloid  SR  has  to  CO  the  semi-diameter  of  the  interior  globe. 

PROPOSITION  LI.     THEOREM  XVIII. 

If  a  centripetal  force  tending  on  all  sides  to  the  centre  C  of  a  globe,  be  in 
all  places  as  the  distance  of  the  place  from  the  centre,  and  by  this  force 
alone  acting  upon  it,  the  body  T  oscillate  (in  the  manner  above  de 
scribed]  in  the  perimeter  of  the  cycloid  QRS ;  /  say,  that  all  the  oscil 
lations,  how  unequal  soever  in  tfiemselves,  will  be  performed  in  equal 
times. 

For  upon  the  tangent  T  W  infinitely  produced  let  fall  the  perpendicular 
CX,  and  join  CT.  Because  the  centripetal  force  with  which  the  body  T 
is  impelled  towards  C  is  as  the  distance  CT,  let  this  (by  Cor.  2,  of  the 
I  ,aws)  be  resolved  into  the  parts  CX,  TX,  of  which  CX  impelling  the 
body  directly  from  P  stretches  the  thread  PT,  and  by  the  resistance  the 
'rhread  makes  to  it  is  totally  employed,  producing  no  other  effect ;  but  the 
3ther  part  TX,  impelling  the  body  transversely  or  towards  X,  directly 
accelerates  the  motion  in  the  cycloid.  Then  it  is  plain  that  the  accelera 
tion  of  the  body,  proportional  to  this  accelerating  force,  will  bo  every 


188 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK  1 


moment  as  the  length  TX,  that  is  (because  CV\ 
WV,  and  TX,  TW  proportional  to  them  are  given), 
as  the  length  TW,  that  is  (by  Cor.  1,  Prop.  XLIX) 
as  the  length  of  the  arc  of  the  cycloid  TR.  If  there 
fore  two  pendulums  APT,  Apt,  be  unequally  drawn 
aside  from  the  perpendicular  AR,  and  let  fall  together, 
their  accelerations  will  be  always  as  the  arcs  to  be  de 
scribed  TR,  tR.  But  the  parts  described  at  the 
beginning  of  the  motion  are  as  the  accelerations,  thai 
is,  as  the  wholes  that  are  to  be  described  at  the  be 
ginning,  and  therefore  the  parts  which  remain  to  be 
described,  and  the  subsequent  accelerations  proportional  to  those  parts,  are 
also  as  the  wholes,  and  so  on.  Therefore  the  accelerations,  and  consequently 
the  velocities  generated,  and  the  parts  described  with  those  velocities,  and 
the  parts  to  be  described,  are  always  as  the  wholes  ;  and  therefore  the  parts 
to  be  described  preserving  a  given  ratio  to  each  other  will  vanish  together, 
that  is,  the  two  bodies  oscillating  will  arrive  together  at  the  perpendicular  AR. 
And  since  on  the  other  hand  the  ascent  of  the  pendulums  from  the  lowest  place 
R  through  the  same  cycloidal  arcs  with  a  retrograde  motion,  is  retarded  in 
the  several  places  they  pass  through  by  the  same  forces  by  which  their  de 
scent  was  accelerated  :  it  is  plain  that  the  velocities  of  their  ascent  and  de 
scent  through  the  same  arcs  are  equal,  and  consequently  performed  in  equal 
times  ;  and,  therefore,  since  the  two  parts  of  the  cycloid  RS  and  RQ  lying 
on  either  side  of  the  perpendicular  are  similar  and  equal,  the  two  pendu 
lums  will  perform  as  well  the  wholes  as  the  halves  of  their  oscillations  in 
the  same  times.  Q.E.D. 

'  COR.  The  force  with  which  the  body  T  is  accelerated  or  retarded  in  any 
place  T  of  the  cycloid,  is  to  the  whole  weight  of  the  same  body  in  the 
highest  place  S  or  Q,  as  the  arc  of  the  cycloid  TR  is  to  the  arc  SR  or  QR 

PROPOSITION  LIL     PROBLEM  XXXIV. 

To  define  the  velocities  of  the  pendulums  in  the  several  places,  and  the 
times  in  which  both  the  entire  oscillations,  and  the  several  parts  of  them 
are  performed. 

About  any  centre  G,  with  the  interval  GH  equal  to 
the  arc  of  the  cycloid  RS,  describe  a  semi-circle  HKM 
bisected  by  the  semi-diameter  GK.  And  if  a  centripe 
tal  force  proportional  to  the  distance  of  the  places  from 
the  centre  tend  to  the  centre  G,  and  it  be  in  the  peri 
meter  HIK  equal  to  the  centripetal  force  in  the  perime 
ter  of  the  globe  Q,OS  tending  towards  its  centre,  and  at 
the  same  time  that  the  pendulum  T  is  let  fall  from  the 
highest  place  S,  a  body,  as  L,  is  let  fall  from  H  to  G  ;  then  because  th< 


SEC.    X.J  OF    NATURAL    PHILOSOPHY.  189 

forces  which  act  upon  the  bodies  are  equal  at  the  be 
ginning,  and  always  proportional  to  the  spaces  to  be 
described  TR,  LG,  and  therefore  if  TR  and  LG  are 
equal,  are  also  equal  in  the  places  T  and  L,  it  is  plain 
that  those  bodies  describe  at  the  beginning  equal  spaces  M 
ST,  HL,  and  therefore  are  still  acted  upon  equally,  and  continue  to  describe 
equal  spaces.  Therefore  by  Prop.  XXXVIII,  the  time  in  which  the  body 
describes  the  arc  ST  is  to  the  time  of  one  oscillation,  as  the  arc  HI  the  time 
in  which  the  body  H  arrives  at  L,  to  the  semi-periphery  HKM,  the  time 
in  which  the  body  H  will  come  to  M.  And  the  velocity  of  the  pendulous 
body  in  the  place  T  is  to  its  velocity  in  the  lowest  place  R,  that  is,  the 
velocity  of  the  body  H  in  the  place  L  to  its  velocity  in  the  place  G,  or  the 
momentary  increment  of  the  line  HL  to  the  momentary  increment  of  the 
line  HG  (the  arcs  HI,  HK  increasing  with  an  equable  flux)  as  the  ordinato 
LI  to  the  radius  GK.  or  as  v/SR2  —  Til2  to  SR.  Hence,  since  in  unequal 
oscillations  there  are  described  in  equal  time  arcs  proportional  to  the  en 
tire  arcs  of  the  oscillations,  there  are  obtained  from  the  times  given,  both 
the  velocities  and  the  arcs  described  in  all  the  oscillations  universally. 
Which  was  first  required. 

Let  now  any  pendulous  bodies  oscillate  in  different  cycloids  described 
within  different  globes,  whose  absolute  forces  are  also  different ;  and  if  the 
absolute  force  of  any  globe  Q.OS  be  called  V,  the  accelerative  force  with 
which  the  pendulum  is  acted  on  in  the  circumference  of  this  globe,  when  it 
begins  to  move  directly  towards  its  centre,  will  be  as  the  distance  of  the 
pendulous  body  from  that  centre  and  the  absolute  force  of  the  globe  con- 
junctly,  that  is,  as  CO  X  V.  Therefore  the  lineola  HY,  which  is  as  this 
accelerated  force  CO  X  V,  will  be  described  in  a  given  time :  and  if  there 
be  erected  the  perpendicular  YZ  meeting  the  circumference  in  Z,  the  nascent 
arc  HZ  will  denote  that  given  time.  But  that  nascent  arc  HZ  is  in  the 
subduplicate  ratio  of  the  rectangle  GHY,  and  therefore  as  v/GH  X  CO  X  V 
Whence  the  time  of  an  entire  oscillation  in  the  cycloid  Q,RS  (it  being  as 
the  semi-periphery  HKM,  wrhich  denotes  that  entire  oscillation,  directly  : 
and  as  the  arc  HZ  which  in  like  manner  denotes  a  given  time  inversely) 
will  be  as  GH  directly  and  v/GH  X  CO  X  V  inversely ;  that  is,  because 

GH  and  SR  are  equal,  as  Vnr,  .    ™  or  (by  Cor.  Prop.  L,)  as  X/-TTVT-  — 

UU  X  V  AO  X   V 

Therefore  the  oscillations  in  all  globes  and  cycloids,  performed  with  what 
absolute  forces  soever,  are  in  a  ratio  compounded  of  the  subduplicate  ratio  of 
the  length  of  the  string  directly,  and  the  subduplicate  ratio  of  the  distance 
between  the  point  of  suspension  and  the  centre  of  the  globe  inversely,  and 
the  subduplicate  ratio  of  the  absolute  force  of  the  globe  inversely  also 
Q.E.I. 


t90  THE    MATHEMATICAL    PRINCIPLES  [Bo^K    1. 

COR.  1.  Hence  also  the  times  of  oscillating,  falling,  and  revolving  bodies 
may  be  compared  among  themselves.  For  if  the  diameter  of  the  wheel 
with  which  the  cycloid  is  described  within  the  globe  is  supposed  equal  to 
the  semi-diameter  of  the  globe,  the  cycloid  will  become  a  right  line  passing 
through  the  centre  of  the  globe,  and  the  oscillation  will  be  changed  into  a 
descent  and  subsequent  ascent  in  that  right  line.  Whence  there  is  given 
both  the  time  of  the  descent  from  any  place  to  the  centre,  and  the  time  equal 
to  it  in  which  the  body  revolving  uniformly  about  the  centre  of  the  globe 
at  any  distance  describes  an  arc  of  a  quadrant  For  this  time  (by 
Case  2)  is  to  the  time  of  half  the  oscillation  in  any  cycloid  QJR.S  as  1  to 

AR 
V  AC' 

COR.  2.  Hence  also  follow  what  Sir  Christopher  Wren  and  M.  Huygevs 
have  discovered  concerning  the  vulgar  cycloid.  For  if  the  diameter  of  the 
globe  be  infinitely  increased,  its  sphacrical  superficies  will  be  changed  into  a 
plane,  and  the  centripetal  force  will  act  uniformly  in  the  direction  of  lines 
perpendicular  to  that  plane,  and  this  cycloid  of  our's  will  become  the  same 
with  the  common  cycloid.  But  in  that  case  the  length  of  the  arc  of  the 
cycloid  between  that  plane  and  the  describing  point  will  become  equal  to 
four  times  the  versed  sine  of  half  the  arc  of  the  wheel  between  the  same 
plane  and  the  describing  point,  as  was  discovered  by  Sir  Christopher  Wren. 
And  a  pendulum  between  two  such  cycloids  will  oscillate  in  a  similar  and 
equal  cycloid  in  equal  times,  as  M.  Huygens  demonstrated.  The  descent 
of  heavy  bodies  also  in  the  time  of  one  oscillation  will  be  the  same  as  M. 
Huygens  exhibited. 

The  propositions  here  demonstrated  are  adapted  to  the  true  constitution 
of  the  Earth,  in  so  far  as  wheels  moving  in  any  of  its  great  circles  will  de 
scribe,  by  the  motions  of  nails  fixed  in  their  perimeters,  cycloids  without  the 
globe ;  and  pendulums,  in  mines  and  deep  caverns  of  the  Earth,  must  oscil 
late  in  cycloids  within  the  globe,  that  those  oscillations  may  be  performed 
in  equal  times.  For  gravity  (as  will  be  shewn  in  the  third  book)  decreases 
in  its  progress  from  the  superficies  of  the  Earth ;  upwards  in  a  duplicate 
ratio  of  the  distances  from  the  centre  of  the  Earth  ;  downwards  in  a  sim 
ple  ratio  of  the  same. 

PROPOSITION  LIII.     PROBLEM  XXXV. 

Granting  the  quadratures  of  curvilinear  figures,  it  is  required  to  find 
the  forces  with  which  bodies  moving  in  given  curve  lines  may  always 
perform  their  oscillations  in  equal  times. 

Let  the  body  T  oscillate  in  any  curve  line  STRQ,,  whose  axis  is  AR 
passing  through  the  centre  of  force  C.  Draw  TX  touching  that  curve  in 
any  place  of  the  body  T,  and  in  that  tangent  TX  take  TY  equal  to  the 
arc  TR.  The  length  of  that  arc  is  known  from  the  common  methods  used 


SEC.  X. 


OF    NATURAL    PHILOSOPHY. 


191 


for  the  quadratures  of  figures.  From  the  point  Y 
draw  the  right  line  YZ  perpendicular  to  the  tangent. 
Draw  CT  meeting  that  perpendicular  in  Z,  and  the 
centripetal  force  will  be  proportional  to  the  right  line 
TZ.  Q.E.I. 

For  if  the  force  with  which  the  body  is  attracted 
from  T  towards  C  be  expressed  by  the  right  line  TZ 
taken  proportional  to  it,  that  force  will  be  resolved 
into  two  forces  TY,  YZ,  of  which  YZ  drawing  the 
body  in  the  direction  of  the  length  of  the  thread  PT, 
docs  not  at  all  change  its  motion  ;  whereas  the  other 
force  TY  directly  accelerates  or  retards  its  mction  in  the  curve  STRQ. 
Wherefore  since  that  force  is  as  the  space  to  be  described  TR,  the  acceler 
ations  or  retardations  of  the  body  in  describing  two  proportional  parts  (u 
greater  arid  a  less)  of  two  oscillations,  will  be  always  as  those  parts,  and 
therefore  will  cause  those  parts  to  be  described  together.  But  bodies  which 
continually  describe  together  parts  proportional  to  the  wholes,  will  describe 
the  wholes  together  also.  Q,.E.l). 

COR.  1.  Hence  if  the  body  T,  hanging  by  a  rectilinear  thread 
AT  from  the  centre  A,  describe  the  circular  arc  STRQ,, 
and  in  the  mean  time  be  acted  on  by  any  force  tending 
downwards  with  parallel  directions,  which  is  to  the  uni 
form  force  of  gravity  as  the  arc  TR  to  its  sine  TN,  the 
times  of  the  several  oscillations  will  be  equal.  For  because 
TZ,  AR  are  parallel,  the  triangles  ATN,  ZTY  are  similar ;  and  there 
fore  TZ  will  be  to  AT  as  TY  to  TN ;  that  is,  if  the  uniform  force  of 
gravity  be  expressed  by  the  given  length  AT,  the  force  TZ.  by  which  the 
oscillations  become  isochronous,  will  be  to  the  force  of  gravity  AT,  as  the 
arc  TR  equal  to  TY  is  to  TN  the  sine  of  that  arc. 

COR.  2.  And  therefore  in  clocks,  if  forces  were  impressed  by  some  ma 
chine  upon  the  pendulum  which  preserves  the  motion,  and  so  compounded 
with  the  force  of  gravity  that  the  whole  force  tending  downwards  should 
be  always  as  a  line  produced  by  applying  the  rectangle  under  the  arc  TR 
and  the  radius  AR  to  the  sine  TN,  all  the  oscillations  will  become 
isochronous. 

PROPOSITION  LIV.    PROBLEM  XXX YI. 

Granting  the  quadratures  of  curvilinear  figures,  it  is  required  to  find 
the  times  in  which  bodies  by  means  of  any  centripetal  force  will  descend 
or  ascend  in  any  curve  lines  described  in,  a  plane  passing  through  the 
centre  of  force. 
Let  the  body  descend  from  any  place  S,  and  move  in  any  curve  ST/R 

given  in  a  plane  passing  through  the  centre  of  force  C.     Join  CS,  and  lei 


192 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK  1 


Q  it  be  divided  into  innumerable  equal  parts,  and  let 
Dd  be  one  of  those  parts.  From  the  centre  C,  with 
the  intervals  CD,  Cd,  let  the  circles  DT,  dt  be  de 
scribed,  meeting  the  curve  line  ST*R  in  T  and  t. 
And  because  the  law  of  centripetal  force  is  given. 
and  also  the  altitude  CS  from  which  the  body  at 
first  fell,  there  will  be  given  the  velocity  of  the  body 
in  any  other  altitude  CT  (by  Prop.  XXXIX).  But 
the  time  in  which  the  body  describes  the  lineola  Tt 
is  as  the  length  of  that  lineola,  that  is,  as  the  secant 
of  the  angle  /TC  directly,  and  the  velocity  inversely. 
Lei,  the  ordinate  DN,  proportional  to  this  time,  be  made  perpendicular  to 
the  right  line  CS  at  the  point  D,  and  because  Dd  is  given,  the  rectangle 
Dd  X  DN,  that  is,  the  area  DNwc?,  will  be  proportional  to  the  same  time. 
Therefore  if  PN/?,  be  a  curve  line  in  which  the  point  N  is  perpetually  found, 
and  its  asymptote  be  the  right  line  SQ,  standing  upon  the  line  CS  at  right 
angles,  the  area  SQPJN  D  will  be  proportional  to  the  time  in  which  the  body 
in  its  descent  hath  described  the  line  ST ;  and  therefore  that  area  being 
found,  the  time  is  also  given.  Q.E.I. 


PROPOSITION  LV.     THEOREM  XIX. 

If  a  body  move  in  any  curve  superficies,  whose  axis  passes  through  the 
centre  of  force,  and  from  the  body  a  perpendicular  be  let  fall  iipon  the 
axis  \  and  a  line  parallel  and  equal  thereto  be  drawn  from  any  given 
point  of  the  axis  ;  I  say,  that  this  parallel  line  will  describe  an  area 
proportional  to  the  time. 

Let  BKL  be  a  curve  superficies,  T  a  body 
revolving  in  it,  STR  a  trajectory  which  the 
body  describes  in  the  same,  S  the  beginning 
of  the  trajectory,  OMK  the  axis  of  the  curve 
superficies,  TN  a  right  line  let  fall  perpendic 
ularly  from  the  body  to  the  axis ;  OP  a  line 
parallel  and  equal  thereto  drawn  from  the 
given  point  O  in  the  axis ;  AP  the  orthogra 
phic  projection  of  the  trajectory  described  by 
the  point  P  in  the  plane  AOP  in  which  the 
revolving  line  OP  is  found :  A  the  beginning 
of  that  projection,  answering  to  the  point  S ; 
TO  a  right  line  drawn  from  the  body  to  the  centre ;  TG  a  part  thereof 
proportional  to  the  centripetal  force  with  which  the  body  tends  towards  the 
centre  C ;  TM  a  right  line  perpendicular  to  the  curve  superficies ;  TI  a 
part  thereof  proportional  to  the  force  of  pressure  with  which  the  body  urges 


SEC.  X.] 


OF    NATURAL    PHILOSOPHY. 


193 


the  superficies,  and  therefore  with  which  it  is  again  repelled  by  the  super 
ficies  towards  M  ;  PTF  a  right  line  parallel  to  the  axis  and  passing  through 
the  body,  and  OF,  IH  right  lines  let  fall  perpendicularly  from  the  points 
G  and  I  upon  that  parallel  PHTF.  I  say,  now.  that  the  area  AGP,  de 
scribed  by  the  radius  OP  from  the  beginning  of  the  motion,  is  proportional 
to  the  time.  For  the  force  TG  (by  Cor.  2,  of  the  Laws  of  Motion)  is  re 
solved  into  the  forces  TF,  FG ;  and  the  force  TI  into  the  forces  TH,  HI ; 
but  the  forces  TF,  TH,  acting  in  the  direction  of  the  line  PF  perpendicular 
to  the  plane  AOP,  introduce  no  change  in  the  motion  of  the  body  but  in  a  di 
rection  perpendicular  to  that  plane.  Therefore  its  motion,  so  far  as  it  has 
the  same  direction  with  the  position  of  the  plane,  that  is,  the  motion  of  the 
point  P,  by  which  the  projection  AP  of  the  trajectory  is  described  in  that 
plane,  is  the  same  as  if  the  forces  TF,  TH  were  taken  away,  and  the  body 
wei  e  acted  on  by  the  forces  FG,  HI  alone ;  that  is,  the  same  as  ,f  the  body 
were  to  describe  in  the  plane  AOP  the  curve  AP  by  means  of  a  centripetal 
force  tending  to  the  centre  O,  and  equal  to  the  sum  of  the  forces  FG  and 
HI.  But  with  such  a  force  as  that  (by  Prop.  1)  the  area  AOP  will  be  de 
scribed  proportional  to  the  time.  Q.E.D. 

COR.  By  the  same  reasoning,  if  a  body,  acted  on  by  forces  tending  to 
two  or  more  centres  in  any  the  same  right  line  CO,  should  describe  in  a 
free  space  any  curve  line  ST,  the  area  AOP  would  be  always  proportional 
to  the  time. 

PROPOSITION  LVI.     PROBLEM  XXXVII. 

Granting  the  quadratures  of  curvilinear  figures,  and  supposing  that 
there  are  given  both  the  law  of  centripetal  force  tending  to  a  given  cen 
tre,  and  the  curve  superficies  whose  axis  passes  through  that  centre ; 
it  is  required  to  find  the  trajectory  which  a  body  will  describe  in  that 
superficies,  when  going  off  from  a  given  place  with  a  given  velocity, 
and  in  a  given  direction  in  that  superficies. 
The  last  construction  remaining,  let  the 
body  T  go  from  the  given  place  S,  in  the  di 
rection  of  a  line  given  by  position,  and  turn 
into  the  trajectory  sought  STR,  whose  ortho 
graphic  projection  in  the  plane  BDO  is  AP. 
And  from  the  given  velocity  of  the  body  in 
the  altitude  SC,  its  velocity  in  any  other  al 
titude  TC  will  be  also  given.     With  that 
velocity,  in  a  given  moment  of  time,  let  the 
body  describe  the  particle  Tt  of  its  trajectory, 
and  let  P/?  be  the  projection  of  that  particle 
described  in  the  plane  AOP.     Join  Op,  and 
a  little  circle  being  described  upon  the  curve  superficies  about  the  centre  T 

13 


194  THE    MATHEMATICAL    PRINCIPLES  [BOOK    I 

with  the  interval  TV  let  the  projection  of  that  little  circle  in  the  plane  AOP 
be  the  ellipsis  pQ.  And  because  the  magnitude  of  that  little  circle  T/,  and 
TN  or  PO  its  distance  from  the  axis  CO  is  also  given,  the  ellipsis  pQ,  will 
be  given  both  in  kind  and  magnitude,  as  also  its  position  to  the  right  line 
PO.  And  since  the  area  PO/?  is  proportional  to  the  time,  and  therefore 
given  because  the  time  is  given,  the  angle  POp  will  be  given.  And  thence 
will  be  given  jo  the  common  intersection  of  the  ellipsis  and.  the  right  line 
Op,  together  with  the  angle  OPp,  in  which  the  projection  APp  of  the  tra 
jectory  cuts  the  line  OP.  But  from  thence  (by  conferring  Prop.  XLI,  with 
Us  2d  Cor.)  the  mariner  of  determining  the  curve  APp  easily  appears. 
Then  from  the  several  points  P  of  that  projection  erecting  to  the  plane 
AOP,  the  perpendiculars  PT  meeting  the  curve  superficies  in  T,  there  will 
be  £iven  the  several  points  T  of  the  trajectory.  Q.E.I. 


SECTION  XL 

f  'f  the  motions  of  bodies  tending'  to  each  other  with  centripetal  forces. 
I  have  hitherto  been  treating  of  the  attractions  of  bodies  towards  an  im 
movable  centre;  though  very  probably  there  is  no  such  thing  existent  in 
nature.  For  attractions  are  made  towards  bodies,  and  the  actions  of  the 
bodies  attracted  and  attracting  are  always  reciprocal  and  equal,  by  Law  III ; 
BO  that  if  there  are  two  bodies,  neither  the  attracted  nor  the  attracting  body 
is  truly  at  rest,  but  both  (by  Cor.  4,  of  the  Laws  of  Motion),  being  as  it 
were  mutually  attracted,  revolve  about  a  common  centre  of  gravity.  And 
if  there  be  more  bodies,  which  are  either  attracted  by  one  single  one  which 
is  attracted  by  them  again,  or  which  all  of  them,  attract  each  other  mutu 
ally  ,  these  bodies  will  be  so  moved  among  themselves,  as  that  their  common 
centre  of  gravity  will  either  be  at  rest,  or  move  uniformly  forward  in  a 
right  line.  I  shall  therefore  at  present  go  on  to  treat  of  the  motion  of 
bodies  mutually  attracting  each  other ;  considering  the  centripetal  forces 
as  attractions  ;  though  perhaps  in  a  physical  strictness  they  may  more  truly 
be  called  impulses.  But  these  propositions  are  to  be  considered  as  purely 
mathematical;  and  therefore,  laying  aside  all  physical  considerations,  I 
make  use  of  a  familiar  way  of  speaking,  to  make  myself  the  more  easily 
understood  by  a  mathematical  reader. 

PROPOSITION  LVII.     THEOREM  XX. 

Two  bodies  attracting  each  other  mutually  describe  similar  figures  about 

their  common  centre  of  gravity,  and  about  each  other  mutually. 

For  the  distances  of  the  bodies  from  their  common  centre  of  gravity  are 

leciprocally  as  the  bodies;  and  therefore  in  a  given   ratio  to  each  other: 

*nd  thence,  bv  composition  of  ratios,  in  a  given  ratio  to  the  whole  distance 


SEC.   XL]  OF    NATURAL    PHILOSOPHY.  195 

between  the  bodies.  Now  these  distances  revolve  about  their  common  term 
with  an  equable  angular  motion,  because  lying  in  the  same  right  line  they 
never  change  their  inclination  to  each  other  mutually  But  right  lines 
that  are  in  a  given  ratio  to  each  other,  and  revolve  about  their  terms  with 
an  equal  angular  motion,  describe  upon  planes,  which  either  rest  with 
those  terms,  or  move  with  any  motion  not  angular,  figures  entirely  similar 
round  those  terms.  Therefore  the  figures  described  by  the  revolution  ot 
these  distances  are  similar.  Q.E.D. 


PROPOSITION  LVIll.     THEOREM  XXI. 

If  two  bodies  attract  each  other  mutually  with  forces  of  any  kind,  and 
in  the  mean  time  revolve  about  the  common  centre  of  gravity  ;  I  say, 
that,  by  the  same  forces,  there  may  be  described  round  either  body  un 
moved  ajigure  similar  and  equal  to  the  figures  ivhich  the  bodies  so 
moving  describe  round  each  other  mutually. 
Let  the  bodies  S  and  P  revolve  about  their  common  centre  of  gravity 

C,  proceeding  from  S  to  T,  and  from  P  to  Q,.     From  the  given  point  s  lot 


there  be  continually  drawn  sp,  sq,  equal  and  parallel  to  SP,  TQ, ;  and  the 
;urve pqv,  which  the  point  p  describes  in  its  revolution  round  the  immovable 
point  s,  will  be  similar  and  equal  to  the  curves  which  the  bodies  S  and  P' 
describe  about  each  other  mutually ;  and  therefore,  by  Theor.  XX,  similar 
to  the  curves  ST  and  PQ,V  which  the  same  bodies  describe  about  their 
common  centre  of  gravity  C  •  and  that  because  the  proportions  of  the  lines 
SC.  CP,  and  SP  or  sp,  to  each  other,  are  given. 

CASE  1.  The  common  centre  of  gravity  C  (by  Cor.  4,  of  the  Laws  of  Mo 
tion)  is  either  at  rest,  or  moves  uniformly  in  a  right  line.  Let  us  first 
suppose  it  at  rest,  and  in  s  and  p  let  there  be  placed  two  bodies,  one  im 
movable  in  s,  the  other  movable  in  p,  similar  and  equal  to  the  bodies  S  arid 
P.  Then  let  the  right  lines  PR  and  pr  touch  the  curves  PQ,  and  pq  ki  P 
and  p,  and  produce  CQ,  and  sq  to  R  and  r.  And  because  the  figures 
CPRQ,  sprq  are  similar,  RQ,  will  be  to  rq  as  CP  to  sp,  and  therefore  in  a 
given  ratio.  Hence  if  the  force  with  which  the  body  P  is  attracted  to 
wards  the  body  S,  and  by  consequence  towards  the  intermediate  point  the 
centre  C,  were  to  the  force  with  which  the  body  p  is  attracted  towards  the 
Centre  5.  in  the  same  given  ratio,  these  forces  would  in  equal  times  attract 


196  THE    MATHEMATICAL    PRINCIPLES  |BoOK   1 

the  bodies  from  the  tangents  PR,  pr  to  the  arcs  PQ,  pq,  through  the  in 
tervals  proportional  to  them  RQ,,  rq  ;  and  therefore  this  last  force  (tending 
to  s)  would  make  the  body  p  revolve  in  the  curve  pqv,  which  would  becomr 
similar  to  the  curve  PQV,  in  which  the  first  force  obliges  the  body  P  i( 
revolve ;  and  their  revolutions  would  be  completed  in  the  same  timeg 
But  because  those  forces  are  not  to  each  other  in  the  ratio  of  CP  to  sp,  bu; 
(by  reason  of  the  similarity  and  equality  of  the  bodies  S  and  s,  P  and  / 
and  the  equality  of  the  distances  SP,  sp)  mutually  equal,  the  bodies  ii 
equal  times  will  be  equally  drawn  from  the  tangents;  and  therefore  tLV 
the  body  p  may  be  attracted  through  the  greater  interval  rq,  there  is  re 
quired  a  greater  time,  which  will  be  in  the  subduplicate  ratio  of  the  inter 
vals  ;  because,  by  Lemma  X,  the  spaces  described  at  the  very  beginning  ol 
the  motion  are  in  a  duplicate  ratio  of  the  times.  Suppose,  then  the  velocity 
of  the  body  p  to  be  to  the  velocity  of  the  body  P  in  a  subduplicate  ratio  of 
the  distance  sp  to  the  distance  CP,  so  that  the  arcs  pq,  PQ,  which  are  in  a 
simple  proportion  to  each  other,  may  be  described  in  times  that  are  in  n 
subduplicate  ratio  of  the  distances  ;  and  the  bodies  P,  p,  always  attracted 
by  equal  forces,  will  describe  round  the  quiescent  centres  C  and  s  similar 
figures  PQV,  pqv,  the  latter  of  which  pqv  is  similar  and  equal  to  the  figure 
ivhich  the  body  P  describes  round  the  movable  body  S.  Q.E.I) 

CASE  2.  Suppose  now  that  the  common  centre  of  gravity,  together  with 
the  space  in  which  the  bodies  are  moved  among  themselves,  proceeds  uni 
formly  in  a  right  line  ;  and  (by  Cor.  6,  of  the  Laws  of  Motion)  all  the  mo 
tions  in  this  space  will  be  performed  in  the  same  manner  as  before ;  and 
therefore  the  bodies  will  describe  mutually  about  each  other  the  same  fig 
ures  as  before,  which  will  be  therefore  similar  and  equal  to  the  figure  pqv. 
Q.E.D. 

COR.  1.  Hence  two  bodies  attracting  each  other  with  forces  proportional 
to  their  distance,  describe  (by  Prop.  X)  both  round  their  common  centre  ol 
gravity,  and  round  each  other  mutually  concentrical  ellipses  ;  and,  vice 
versa,  if  such  figures  are  described,  the  forces  are  proportional  to  the  dis 
tances. 

COR.  2.  And  two  bodies,  whose  forces  are  reciprocally  proportional  to 
the  square  of  their  distance,  describe  (by  Prop.  XI,  XII,  XIII),  both  round 
their  common  centre  of  gravity,  and  round  each  other  mutually,  conic  sec 
tions  having  their  focus  in  the  centre  about  which  the  figures  are  described. 
And,  vice  versa,  if  such  figures  are  described,  the  centripetal  forces  are  re 
ciprocally  proportional  to  the  squares  of  the  distance. 

COR.  3.  Any  two  bodies  revolving  round  their  common  centre  of  gravity 
describe  areas  proportional  to  the  times,  by  radii  drawn  both  to  that  centre 
and  to  each  other  mutually- 


>EC.    XL]  OF    NATURAL    PHILOSOPHY.  197 

PROPOSITION  LIX.     THEOREM  XXII. 

The  periodic  time  of  two  bodies  S  and  P  revolving  round  their  common 
centre  of  gravity  C,is  to  the  periodic  time  of  one  of  the  bwlies  1?  re 
volving  round  the  other  S  remaining  unmoved,  and  describing  a  fig 
ure  similar  and  equal  to  those  which  the  bodies  describe  about  each 
other  mutually r,  in  a  subduplicate  ratio  of  the  other  body  S  to  the  sii/rn 
of  the  bodies  S  -f  P. 

For,  by  the  demonstration  of  the  last  Proposition,  the  times  in  which 
any  similar  arcs  PQ  and  pq  are  described  are  in  a  subduplicate  ratio  of  the 
distances  CP  and  SP,  or  sp,  that  is,  in  a  subduplicate  ratio  of  the  ody  S 
to  the  sum  of  the  bodies  S  +  P.  And  by  composition  of  ratios,  the  sums 
of  the  times  in  which  all  the  similar  arcs  PQ  and  pq  are  described,  that  is, 
the  whole  times  in  which  the  whole  similar  figures  are  described  are  in  the 
same  subduplicate  ratio.  Q.E.D. 

PROPOSITION  LX.     THEOREM  XXIII. 

If  two  bodies  S  and  P,  attracting  each  other  with  forces  reciprocally  pro 
portional  to  the  squares  of  their  distance,  revolve  about  their  common 
centre  of  gravity  ;  I  say,  that  the  principal  axis  of  the  ellipsis  which 
either  of  the  bodies,  as  P,  describes  by  this  motion  about  the  other  S, 
will  be  to  the  principal  axis  of  the  ellipsis,  which  the  same  body  P  may 
describe  in  the  same  periodical  time  about  the  other  body  S  quiescent, 
as  the  sum  of  the  two  bodies  S  +  P  to  the  first  of  two  mean,  propor 
tionals  between  that  sum  and  the  other  body  S. 

For  if  the  ellipses  described  were  equal  to  each  other,  their  periodic  times 
by  the  last  Theorem  would  be  in  a  subduplicate  ratio  of  the  body  S  to  the 
sum  of  the  bodies  S  4-  P.  Let  the  periodic  time  in  the  latter  ellipsis  be 
diminished  in  that  ratio,  and  the  periodic  times  will  become  equal ;  but, 
by  Prop.  XV,  the  principal  axis  of  the  ellipsis  will  be  diminished  in  a  ratio 
sesquiplicate  to  the  former  ratio  ;  that  is,  in  a  ratio  to  which  the  ratio  of 
S  to  S  4-  P  is  triplicate ;  and  therefore  that  axis  will  be  to  the  principal 
axis  of  the  other  ellipsis  as  the  first  of  two  mean  proportionals  between  S 
-f-  P  and  S  to  S  4-  P.  And  inversely  the  principal  axis  of  the  ellipsis  de 
scribed  about  the  movable  body  will  be  to  the  principal  axis  of  that  described 
round  the  immovable  as  S  +  P  to  the  first  of  two  mean  proportionals  be 
tween  S  4-  P  and  S.  Q.E.D. 

PROPOSITION  LXI.    THEOREM  XXIV. 

If  two  bodies  attracting  each  other  with  any  kind  of  forces,  and  not 
otherwise  agitated  or  obstructed,  are  moved  in  any  manner  whatsoever, 
those  'motions  will  be  the  same  as  if  they  did  not  at  all  attract  each 
other  mutually,  but  were  both  attracted  with  the  same  forces  by  a  third 
body  placed  in  their  common  centre  of  gravity  ;  and  the  law  of  the 


198  THE    MATHEMATICAL    PRINCIPLES  [BOOK    I 

attracting  Jones  will  be  the  sam#  in  respect  of  the  distance  of  the. 

bodies  from,  the  common  centre,  as  in  respect  of  the  distance  between 

the  two  bodies. 

For  those  forces  with  which  the  bodies  attract  each  other  mutually,  by 
tending  to  the  bodies,  tend  also  to  the  common  centre  of  gravity  lying  di 
rectly  between  them  ;  and  therefore  are  the  same  as  if  they  proceeded  from 
'an  intermediate  body.  QJG.D. 

And  because  there  is  given  the  ratio  of  the  distance  of  either  body  from 
that  common  centre  to  the  distance  between  the  two  bodies,  there  is  given, 
of  course,  the  ratio  of  any  power  of  one  distance  to  the  same  power  of  the 
.  ther  distance ;  and  also  the  ratio  of  any  quantity  derived  in  any  manner 
from  one  of  the  distances  compounded  any  how  with  given  quantities,  to 
another  quantity  derived  in  like  manner  from  the  other  distance,  and  as 
many  given  quantities  having  that  given  ratio  of  the  distances  to  the  first 
Therefore  if  the  force  with  which  one  body  is  attracted  by  another  be  di 
rectly  or  inversely  as  the  distance  of  the  bodies  from  each  other,  or  as  any 
power  of  that  distance  ;  or,  lastly,  as  any  quantity  derived  after  any  man 
ner  from  that  distance  compounded  with  given  q-uantities ;  then  will  the 
same  force  with  which  the  same  body  is  attracted  to  the  common  centre  of 
gravity  be  in  like  manner  directly  or  inversely  as  the  distance  of  the  at 
tracted  body  from  the  common  centre,  or  as  any  power  of  that  distance ; 
or,  lastly,  as  a  quantity  derived  in  like  sort  from  that  distance  compounded 
with  analogous  given  quantities.  That  is,  the  law  of  attracting  force  will 
be  the  same  with  respect  to  both  distances.  Q,.E.D. 

PROPOSITION  LXII.     PROBLEM  XXXVIII. 

To  determine  the  motions  of  two  bodies  which  attract  each  other  with 
forces  reciprocally  proportional  to  the  squares  of  the  distance  between 
them,  aflid  are,  let  fall  from  given  places. 

The  bodies,  by  the  last  Theorem,  will  be  moved  in  the  same  manner  as 
if  they  were  attracted  by  a  third  placed  in  the  common  centre  of  their 
gravity ;  and  by  the  hypothesis  that  centre  will  be  quiescent  at  the  begin 
ning  of  their  motion,  and  therefore  (by  Cor.  4,  of  the  Laws  of  Motion)  will 
be  always  quiescent.  The  motions  of  the  bodies  are  therefore  to  be  deter 
mined  (by  Prob.  XXV)  in  the  same  manner  as  if  they  were  impelled  by 
forces  tending  to  that  centre:  and  then  we  shall  have  the  motions  of  the 
bodies  attracting  each  other  mutually.  Q.E.I. 

PROPOSITION  LXIII.     PROBLEM  XXXIX. 

To  determine  the  motions  of  two  bodies  attracting  each  other  with  forces 
reciprocally  proportional  to  the  squares  of  their  distance,  and  going 
off  from  given  places  in,  given  directions  with  given  velocities. 
The  motions  of  the  bodies  at  the  beginning  being  given,  there  is  given 


SEC.    XL]  OF    NATURAL    PHILOSOPHY.  1  % 

also  the  uniform  motion  of  the  common  centre  of  gravity,  and  the  motion 
of  the  space  which  moves  along  with  this  centre  uniformly  in  a  right  line, 
and  also  the  very  first,  or  beginning  motions  of  the  bodies  in  respect  of  this 
space.  Then  (by  Cor.  5,  of  the  Laws,  and  the  last  Theorem)  the  subse 
quent  motions  will  be  performed  in  the  same  manner  in  that  space,  as  if 
that  space  together  with  the  common  centre  of  gravity  were  at  rest,  and  as 
if  the  bodies  did  not  attract  each  other,  but  were  attracted  by  a  third  body 
placed  in  that  centre.  The  motion  therefore  in  this  movable  space  of  each 
body  going  off  from  a  given  place,  in  a  given  direction,  with  a  given  velo 
city,  and  acted  upon  by  a  centripetal  force  tending  to  that  centre,  is  to  be 
determined  by  Prob.  IX  and  XXVI,  and  at  the  same  time  will  be  obtained 
the  motion  of  the  other  round  the  same  centre.  With  this  motion  com 
pound  the  uniform  progressive  motion  of  the  entire  system  of  the  space  and 
the  bodies  revolving  in  it,  and  there  will  be  obtained  the  absolute  motion 
of  the  bodies  in  immovable  space.  Q..E.I. 

PROPOSITION  LXIV.     PROBLEM  XL. 

Supposing  forces  with  which  bodies  mutually  attract  each  other  to  in 
crease  in  a  simple  ratio  of  their  distances  from  the  centres  ;  it  is  ro- 
quired  to  find  the  motions  of  several  bodies  among  themselves. 
Suppose  the  first  two  bodies  T  and  L 
to  have  their  common  centre  of  gravity  in 
L).     These,  by  Cor.  1,  Theor.  XXI,  will  S 
describe  ellipses  having  their  centres  in  D, 
the   magnitudes    of    which    ellipses   are 

known  by  Prob.  V.  J- -- \- ?  L 

Let  now  a  third  body  S  attract  the  two 
former  T  and  L  with  the  accelerative  forces  ST,  SL,  and  let  it  be  attract 
ed  again  by  them.  The  force  ST  (by  Cor.  2,  of  the  Laws  of  Motion)  is 
resolved  into  the  forces  SD,  DT ;  and  the  force  SL  into  the  forces  SD  and 
DL.  Now  the  forces  DT,  DL.  which  are  as  their  sum  TL,  and  therefore 
as  the  accelerative  forces  with  which  the  bodies  T  and  L  attract  each  other 
mutually,  added  to  the  forces  of  the  bodies  T  and  L,  the  first  to  the  first, 
and  the  last  to  the  last,  compose  forces  proportional  to  the  distances  DT 
and  DL  as  before,  but  only  greater  than  those  former  forces :  and  there 
fore  (by  Cor.  1,  Prop.  X,  and  Cor.  l,and  8,  Prop.  IV)  they  will  cause  those 
bodies  to  describe  ellipses  as  before,  but  with  a  swifter  motion.  The  re 
maining  accelerative  forces  SD  and  DL,  by  the  motive  forces  SD  X  Tand 
SD  X  L,  which  are  as  the  bodies  attracting  those  bodies  equally  and  in  the 
direction  of  the  lines  TI,  LK  parallel  to  DS,  do  not  at  all  change  their  situ 
ations  with  respect  to  one  another,  but  cause  them  equally  to  approach  to 
the  line  IK ;  which  must  be  imagined  drawn  through  the  middle  of  the 
body  S,  and  perpendicular  to  the  line  DS.  But  that  approach  to  the  line 


200  THE    MATHEMATICAL    PRINCIPLES  [BoOK    I. 

IK  will  be  hindered  by  causing  the  system  of  the  bodies  T  and  L  on  one 
side,  and  the  body  S  on  the  other,  with  proper  velocities,  to  revolve  round 
the  common  centre  of  gravity  C.  With  such  a  motion  the  body  S,  because 
the  sum  of  the  motive  forces  SD  X  T  and  SD  X  L  is  proportional  to  the 
distance  OS,  tends  to  the  centre  C,  will  describe  an  ellipsis  round  the  same 
centre  C;  and  the  point  D,  because  the  lines  CS  and  CD  are  proportional, 
will  describe  a  like  ellipsis  over  against  it.  But  the  bodies  T  and  L,  at 
tracted  by  the  motive  forces  SD  X  T  and  SD  X  L,  the  first  by  the  first, 
and  the  last  by  the  last,  equally  and  in  the  direction  of  the  parallel  lines  TI 
and  LK,  as  was  said  before,  will  (by  Cor.  5  and  6,  of  the  Laws  of  Motion) 
continue  to  describe  their  ellipses  round  the  movable  centre  D,  as  before. 
Q.E.I. 

Let  there  be  added  a  fourth  body  V,  and,  by  the  like  reasoning,  it  will 
be  demonstrated  that  this  body  and  the  point  C  will  describe  ellipses  about 
the  common  centre  of  gravity  B ;  the  motions  of  the  bodies  T,  L,  and  S 
round  the  centres  D  and  C  remaining  the  same  as  before ;  but  accelerated. 
Arid  by  the  same  method  one  may  add  yet  more  bodies  at  pleasure.  Q..E.I. 
^This  would  be  the  case,  though  the  bodies  T  and  L  attract  each  other 
mutually  with  accelerative  forces  either  greater  or  less  than  those  with 
which  they  attract  the  other  bodies  in  proportion  to  their  distance.  Let 
all  the  mutual  accelerative  attractions  be  to  each  other  as  the  distances 
multiplied  into  the  attracting  bodies  ;  and  from  what  has  gone  before  it 
will  easily  be  concluded  that  all  the  bodies  will  describe  different  ellipses 
with  equal  periodical  times  about  their  common  centre  of  gravity  B,  in  an 
immovable  plane.  Q.E.I. 

PROPOSITION  LXV.     THEOREM  XXV. 

Bodies,  whose  forces  decrease  in  a  duplicate  ratio  of  their  distances  from 
their  centres,  'may  move  among"  themselves  in  ellipses  ;  and  by  radii 
drawn  to  the  foci  may  describe  areas  proportional  to  the  times  very 
nearly. 

In  the  last  Proposition  we  demonstrated  that  case  in  which  the  motions 
will  be  performed  exactly  in  ellipses.  The  more  distant  the  law  of  the 
forces  is  from  the  law  in  that  case,  the  more  will  the  bodies  disturb  each 
other's  motions ;  neither  is  it  possible  that  bodies  attracting  each  other 
mutually  according  to  the  law  supposed  in  this  Proposition  should  move 
exactly  in  ellipses,  unless  by  keepirg  a  certain  proportion  of  distances  from 
each  other.  However,  in  the  following  crises  the  orbits  will  not  much  dif 
fer  from  ellipses. 

CASE  I.  Imagine  several  lesser  bodies  to  revolve  about  some  very  great 
one  at  different  distances  from  it,  and  suppose  absolute  forces  tending  to 
rvery  one  of  the  bodies  proportional  to  each.  And  because  (by  Cor.  4,  ol 
the  I  aws)  the  common  centre  of  gravity  of  them  all  is  either  at  rest,  01 


iSEC.     XL]  OF    NATURAL    PHILOSOPHY.  20  i 

moves  uniformly  forward  in  a  right  line,  suppose  the  lesser  bodies  so  small 
that  the  groat  body  may  be  never  at  a  sensible  distance  from  that  centre  ; 
and  then  the  great  body  will,  without  any  sensible  error,  be  either  at  rest, 
or  move  uniformly  forward  in  a  right  line;  and  the  lesser  will  revolve 
about  that  great  one  in  ellipses,  and  by  radii  drawn  thereto  will  describe 
areas  proportional  to  the  times ;  if  we  except  the  errors  that  may  be  intro 
duced  by  the  receding  of  the  great  body  from  the  common  centre  of  gravity, 
or  by  the  mutual  actions  of  the  lesser  bodies  upon  each  other.  But  the 
lesser  bodies  may  be  so  far  diminished,  as  that  this  recess  and  the  mutual 
actions  of  the  bodies  on  each  other  may  become  less  than  any  assignable; 
and  therefore  so  as  that  the  orbits  may  become  ellipses,  and  the  areas  an 
swer  to  the  times,  without  any  error  that  is  not  less  than  any  assignable. 
Q.E.O. 

CASE  2.  Let  us  imagine  a  system  of  lesser  bodies  revolving  about  a  very 
great  one  in  the  manner  just  described,  or  any  other  system  of  two  bodies 
revolving  about  each  other  to  be  moving  uniformly  forward  in  a  right  line,  and 
in  the  mean  time  to  be  impelled  sideways  by  the  force  of  another  vastly  greater 
body  situate  at  a  great  distance.  And  because  the  equal  accelerative  forces 
with  which  the  bodies  are  impelled  in  parallel  directions  do  not  change  the 
situation  of  the  bodies  with  respect  to  each  other,  but  only  oblige  the  whole 
system  to  change  its  place  while  the  parts  still  retain  their  motions  among 
themselves,  it  is  manifest  that  no  change  in  those  motions  of  the  attracted 
bodies  can  arise  from  their  attractions  towards  the  greater,  unless  by  the 
inequality  of  the  accelerative  attractions,  or  by  the  inclinations  of  the  lines 
towards  each  other,  in  whose  directions  the  attractions  are  made.  Suppose, 
therefore,  all  the  accelerative  attractions  made  towards  the  great  body 
to  be  among  themselves  as  the  squares  of  the  distances  reciprocally ;  and 
then,  by  increasing  the  distance  of  the  great  body  till  the  differences  of  fhe 
right  lines  drawn  from  that  to  the  others  in  respect  of  their  length,  and  the 
inclinations  of  those  lines  to  each  other,  be  less  than  any  given,  the  mo 
tions  of  the  parts  of  the  system  will  continue  without  errors  that  are  not 
less  than  any  given.  And  because,  by  the  small  distance  of  those  parts  from 
each  other,  the  whole  system  is  attracted  as  if  it  were  but  one  body,  it  will 
therefore  be  moved  by  this  attraction  as  if  it  were  one  body  ;  that  is,  its 
centre  of  gravity  will  describe  about  the  great  bod/  one  of  the  conic  sec 
tions  (that  is,  a  parabola  or  hyperbola  when  the  attraction  is  but  languid 
and  an  ellipsis  when  it  is  more  vigorous) ;  and  by  radii  drawn  thereto,  it 
will  describe  areas  proportional  to  the  times,  without  any  errors  but  thos« 
which  arise  from  the  distances  of  the  parts,  which  are  by  the  supposition 
exceedingly  small,  and  may  be  diminished  at  pleasure.  Q,.E.O. 

By  a  like  reasoning  one  may  proceed  to  more  compounded  cases  in  in- 
finitum. 

COR  1.  In  the  second  Case,  the  nearer  the  very  great  body  approaches  to 


^0'^  THE    MATHEMATICAL    PRINCIPLES  [CoOK    I 

the  system  of  two  or  more  revolving  bodies,  the  greater  will  the  pertur 
bation  be  of  the  motions  of  the  parts  of  the  system  among  themselves;  be 
cause  the  inclinations  of  the  lines  drawn  from  that  great  body  to  those 
parts  become  greater  ;  and  the  inequality  of  the  proportion  is  also  greater. 

COR.  2.  But  the  perturbation  will  be  greatest  of  all,  if  we  suppose  the 
uccelerative  attractions  of  the  parts  of  the  system  towards  the  greatest  body 
of  all  are  not  to  each  other  reciprocally  as  the  squares  of  the  distances 
from  that  great  body ;  especially  if  the  inequality  of  this  proportion  be 
greater  than  the  inequality  of  the  proportion  of  the  distances  from  the 
great  body.  For  if  the  accelerative  force,  acting  in  parallel  directions 
and  equally,  causes  no  perturbation  in  the  motions  of  the  parts  of  the 
system,  it  must  of  course,  when  it  acts  unequally,  cause  a  perturbation  some 
where,  which  will  be  greater  or  less  as  the  inequality  is  greater  or  less. 
The  excess  of  the  greater  impulses  acting  upon  some  bodies,  and  not  acting 
upon  others,  must  necessarily  change  their  situation  among  themselves.  And 
this  perturbation,  added  to  the  perturbation  arising  from  the  inequality 
and  inclination  of  the  lines,  makes  the  whole  perturbation  greater. 

COR.  '*.  Hence  if  the  parts  of  this  system  move  in  ellipses  or  circles 
without  any  remarkable  perturbation,  it  is  manifest  that,  if  they  are  at  all 
impelled  by  accelerative  forces  tending  to  any  other  bodies,  the  impulse  is 
very  weak,  or  else  is  impressed  very  near  equally  and  in  parallel  directions 
upon  all  of  them. 

PROPOSITION  LXVL     THEOREM  XXVI. 

Tf  three  bodies  whose  forces  decrease  in  a  duplicate  ratio  of  the  distances 
attract  each  other  mutually  ;  and  the  accelerative  attractions  of  any 
two  towards  the  third  be  between  themselves  reciprocally  as  the  squares, 
of  the  distances  ;  and  the  two  least  revolve  about  the  greatest ;  I  say, 
that  the  interior  of  the  tivo  revolving  bodies  will,  by  radii  drawn  to  the 
innermost  and  greatest,  describe  round  thai  body  areas  more  propor 
tional  to  the  times,  and  a  figure  more  approaching  to  that  of  an  ellip 
sis  having  its  focus  in  the  point  of  concourse  of  the  radii,  if  that  great 
body  be  agitated  by  those  attractions,  than  it  would  do  if  lhat  great 
body  were  not  attracted  at  all  by  the  lesser,  but  remained  at  rest ;  or 
than  it  would  if  that  great  body  were  very  much  more  or  very  much 
less  attracted,  <>r  very  much  more  or  very  much  less  agitated,  by  the 
attractions. 
This  appears  plainly  enough  from  the  demonstration  of  the  second 

Corollary  of  tl.e  foregoing  Proposition;  but  it  may  be  made  out   after 

this  manner  by  a  way  of  reasoning  more  distinct  and  more  universally 

convincing. 

CASE  1.  Let  the  lesser  bodies  P  and  S  revolve  in  the  same  plane  about 

the  greatest  body  T,  the  body  P  describing  the  interior  orbit  PAB,  and  S 


SEC.    XI.J  OF    NATURAL    PHILOSOPHY.  203 

the  exterior  orbit  ESE.     Let  SK  be  the  mean  distance  of  the  bodies  P  and 

S ;  and  let  the  accelerative  attraction  of  the  body  P  towards  S,  at  that 

mean  distance,  be  expressed  by  that  line  SK.     Make  SL  to   SK  as  the 

E  C 


square  of  SK  to  the  square  of  SP,  and  SL  will  be  the  accelerative  attrac 
tion  of  the  body  P  towards  S  at  any  distance  SP.  Join  PT,  and  draw 
LM  parallel  to  it  meeting  ST  in  M;  and  the  attraction  SL  will  be  resolv 
ed  (by  Cor.  2.  of  the  Laws  of  Motion)  into  the  attractions  SM,  LM.  And 
so  the  body  P  will  be  urged  with  a  threefold  accelerative  force.  One  of 
these  forces  tends  towards  T,  and  arises  from  the  mutual  attraction  of  the 
bodies  T  and  P.  By  this  force  alone  the  body  P  would  describe  round  the 
body  T,  by  the  radius  PT,  areas  proportional  to  the  times,  and  an 
ellipsis  whose  focus  is  in  the  centre  of  the  body  T  ;  and  this  it  would  do 
whether  the  body  T  remained  unmoved,  or  whether  it  were  agitated  by  that 
attraction.  This  appears  from  Prop.  XI,  and  Cor.  2  and  3  of  Theor. 
XXI.  The  other  force  is  that  of  the  attraction  LM,  which,  because  it 
tends  from  P  to  T,  will  be  superadded  to  and  coincide  with  the  former 
force ;  and  cause  the  areas  to  be  still  proportional  to  the  times,  by  Cor.  3, 
Theor.  XXI.  But  because  it  is  not  reciprocally  proportional  to  the  square 
of  the  distance  PT,  it  will  compose,  when  added  to  the  former,  a  force 
varying  from  that  proportion  :  which  variation  will  be  the  greater  by  how 
much  the  proportion  of  this  force  to  the  former  is  greater,  cceteris  paribus. 
Therefore,  since  by  Prop.  XI,  and  by  Cor.  2,  Theor.  XXI,  the  force  with 
which  the  ellipsis  is  described  about  the  focus  T  ought  to  be  directed  to 
that  focus,  and  to  be  reciprocally  proportional  to  the  square  of  the  distance 
PT,  that  compounded  force  varying  from  that  proportion  will  make  the 
orbit  PAB  vary  from  the  figure  of  an  ellipsis  that  has  its  focus  in  the  point 
'I1 ;  and  so  much  the  more  by  how  much  the  variation  from  that  proportion 
is  greater ;  and  by  consequence  by  how  much  the  proportion  of  the  second 
force  LM  to  the  first  force  is  greater,  cceteris  paribus.  But  now  the  third 
force  SM,  attracting  the  body  P  in  a  direction  parallel  to  ST,  composes  with 
the  other  forces  a  new  force  which  is  no  longer  directed  from  P  to  T :  and  which 
varies  so  much  more  from  this  direction  by  how  much  the  proportion  of  this 
third  force  to  the  other  forces  is  greater,  cceteris  paribus  ;  arid  therefore  causes 
the  body  P  to  describe,  by  the  radius  TP,  areas  no  longer  proportional  to  the 
times :  and  therefore  makes  the  variation  from  that  proportionality  so  much 
greater  by  how  much  the  proportion  of  this  force  to  the  others  is  greater. 
But  this  third  force  will  increase  the  variation  of  the  orbit  PAB  from  th* 


THE    MATHEMATICAL    PRINCIPLES  [BOOK    1 

elliptical  figure  before-mentioned  upon  two  accounts ;  first  because  that 
force  is  not  directed  from  P  to  T  ;  and,  secondly,  because  it  is  not  recipro 
cally  proportional  to  the  square  of  the  distance  PT.  These  things  being 
premised,  it  is  manifest  that  the  areas  are  then  most  nearly  proportional  to 
the  times,  when  that  third  force  is  the  least  possible,  the  rest  preserving 
their  former  quantity  ;  and  that  the  orbit  PAB  does  then  approach  nearest 
to  the  elliptical  figure  above-mentioned,  when  both  the  second  and  third, 
but  especially  the  third  force,  is  the  least  possible;  the  first  force  remain 
ing  in  its  former  quantity. 

Let  the  accelerative  attraction  of  the  body  T  towards  S  be  expressed  by 
the  line  SN  ;  then  if  the  accelerative  attractions  SM  and  SN  were  equal, 
these,  attracting  the  bodies  T  and  P  equally  and  in  parallel  directions 
would  not  at  all  change  their  situation  with  respect  to  each  other.  The  mo 
tions  of  the  bodies  between  themselves  would  be  the  same  in  that  case  as  if 
those  attractions  did  not  act  at  all,  by  Cor.  6,  of  the  Laws  of  Motion.  And, 
by  a  like  reasoning,  if  the  attraction  SN  is  less  than  the  attraction  SM,  it 
will  take  away  out  of  the  attraction  SM  the  part  SN,  so  that  there  will  re 
main  only  the  part  (of  the  attraction)  MN  to  disturb  the  proportionality  of 
the  areas  and  times,  and  the  elliptical  figure  of  the  orbit.  And  in  like 
manner  if  the  attraction  SN  be  greater  than  the  attraction  SM,  the  pertur 
bation  of  the  orbit  and  proportion  will  be  produced  by  the  difference  MN 
alone.  After  this  manner  the  attraction  SN  reduces  always  the  attraction 
SM  to  the  attraction  MN,  the  first  and  second  attractions  rema  ning  per 
fectly  unchanged ;  and  therefore  the  areas  and  times  come  then  nearest  to 
proportionality,  and  the  orbit  PAB  to  the  above-mentioned  elliptical  figure, 
when  the  attraction  MN  is  either  none,  or  the  least  that  is  possible;  that 
is,  when  the  accelerative  attractions  of  the  bodies  P  and  T  approach  as  near 
as  possible  to  equality ;  that  is,  when  the  attraction  SN  is  neither  none  at 
all,  nor  less  than  the  least  of  all  the  attractions  SM,  but  is,  as  it  were,  a 
mean  between  the  greatest  and  least  of  all  those  attractions  SM,  that  is, 
not  much  greater  nor  much  less  than  the  attraction  SK.  Q.E.D. 

CASE  2.  Let  now  the  lesser  bodies  P.  S,  revolve  about  a  greater  T  in  dif 
ferent  planes ;  and  the  force  LM,  acting  in  the  direction  of  the  line  PT 
situate  in  the  plane  of  the  orbit  PAB,  will  have  the  same  effect  as  before  ; 
neither  will  it  draw  the  body  P  from  the  plane  of  its  orbit.  But  the  other 
force  NM  acting  in  the  direction  of  a  line  parallel  to  ST  (and  which,  there 
fore,  when  the  body  S  is  without  the  line  of  the  nodes  is  inclined  to  the 
plane  of  the  orbit  PAB),  besides  the  perturbation  of  the  motion  just  now 
spoken  of  as  to  longitude,  introduces  another  perturbation  also  as  to  latitude, 
attracting  the  body  P  out  of  the  plane  of  its  orbit.  And  this  perturbation, 
in  any  given  situation  of  the  bodies  P  and  T  to  each  other,  will  be  as  the 
generating  force  MN ;  and  therefore  becomes  least  when  the  force  MN  is 
least,  that  is  (as  was  just  now  shewn),  where  the  attraction  SN  is  not  nrirb 
greater  nor  much  less  than  the  attraction  SK.  Q.E.D. 


SK-C.  XL]  OF  NATURAL  PHILOSOPHY.  205 

COR.  1.  Hence  it  may  be  easily  collected,  that  if  several  less  bodies  P 
8,  R,  &c.;  revolve  about  a  very  great  body  T,  the  motion  of  the  innermost 
revolving  body  P  will  be  least  disturbed  by  the  attractions  of  the  others. 
when  the  great  body  is  as  well  attracted  and  agitated  by  the  rest  (accord 
ing  to  the  ratio  of  the  accelerative  forces)  as  the  rest  are  by  each  other 
mutually. 

COR.  2.  In  a  system  of  three  bodies,  T,  P,  S,  if  the  accelerative  attrac 
tions  of  any  two  of  them  towards  a  third  be  to  each  other  reciprocally  as  the 
squares  of  the  distances,  the  body  P,  by  the  radius  PT,  will  describe  its  area 
about  the  body  T  swifter  near  the  conjunction  A  and  the  opposition  B  than  it 
will  near  the  quadratures  C  arid  D.  For  every  force  with  which  the  body  P 
is  acted  on  and  the  body  T  is  not,  and  which  does  not  act  in  the  direction  of 
the  line  PT,  does  either  accelerate  or  retard  the  description  of  the  area, 
according  as  it  is  directed,  whether  in  consequentia  or  in  cwtecedentia. 
Such  is  the  force  NM.  This  force  in  the  passage  of  the  body  P  frcm  C 
to  A  is  directed  in  consequentia  to  its  motion,  and  therefore  accelerates 
it;  then  as  far  as  D  in  atttecedentia,  and  retards  the  motion;  then  in,  con 
sequentia  as  far  as  B  ;  and  lastly  in  antecedentia  as  it  moves  from  B  to  C. 

COR.  3.  And  from  the  same  reasoning  it  appears  that  the  body  P  ccBteris 
paribuSj  moves  more  swiftly  in  the  conjunction  and  opposition  than  in  the 
quadratures. 

COR.  4.  The  orbit  of  the  body  P,  cc&teris  paribus,  is  more  curve  at  the 
quadratures  than  at  the  conjunction  and  opposition.  For  the  swifter 
bodies  move,  the  less  they  deflect  from  a  rectilinear  path.  And  besides  the 
force  KL,  or  NM,  at  the  conjunction  and  opposition,  is  contrary  to  the 
force  with  which  the  body  T  attracts  the  body  P,  and  therefore  diminishes 
that  force ;  but  the  body  P  will  deflect  the  less  from  a  rectilinear  path  the 
less  it  is  impelled  towards  the  body  T. 

COR.  5.  Hence  the  body  P,  cceteris  paribus,  goes  farther  from  the  body 
T  at  the  quadratures  than  at  the  conjunction  and  opposition.     This  is  said, 
E  C_  L 


B 

however,  supposing  no  regard  had  to  the  motion  of  eccentricity.  For  if 
the  orbit  of  the  body  P  be  eccentrical,  its  eccentricity  (as  will  be  shewn 
presently  by  Cor.  9)  will  be  greatest  when  the  apsides  are  in  the  syzy- 
gies;  and  thence  it  may  sometimes  come  to  pass  that  the  body  P.  in  its 
near  approach  to  the  farther  apsis,  may  go  farther  from  the  body  T  at  the 
syzygies  than  at  the  quadratures. 

COR.  6.  Because  the  centripetal  force  of  the  central  body  T,  by  which 


206  THE    MATHEMATICAL    PRINCIPLES  [BOOK.    1 

the  body  P  is  retained  in  its  orbit,  is  increased  at  the  quadratures  by  tho 
addition  caused  by  the  force  LM,  and  diminished  at  the  syzygies  by  the 
subduction  caused  by  the  force  KL,  and,  because  the  force  KL  is  greater 
than  LM,  it  is  more  diminished  than  increased ;  and,  moreover,  since  that 
centripetal  force  (by  Cor.  2,  Prop.  IV)  is  in  a  ratio  compounded  of  the  sim 
ple  ratio  of  the  radius  TP  directly,  and  the  duplicate  ratio  of  the  periodi 
cal  time  inversely ;  it  is  plain  that  this  compounded  ratio  is  diminished  by 
the  action  of  the  force  KL ;  and  therefore  that  the  periodical  time,  supposing 
the  radius  of  the  orbit  PT  to  remain  the  same,  will  be  increased,  and  that 
in  the  subduplicate  of  that  ratio  in  which  the  centripetal  force  is  diminish 
ed  ;  and,  therefore,  supposing  this  radius  increased  or  diminished,  the  peri 
odical  time  will  be  increased  more  or  diminished  less  than  in  the  sesquipli- 
cate  ratio  of  this  radius,  by  Cor.  6,  Prop.  IV.  If  that  force  of  the  central 
body  should  gradually  decay,  the  body  P  being  less  and  less  attracted  would 
go  farther  and  farther  from  the  centre  T ;  and,  on  the  contrary,  if  it  were 
increased,  it  would  draw  nearer  to  it.  Therefore  if  the  action  of  the  distant 
body  S,  by  which  that  force  is  diminished,  were  to  increase  and  decrease 
by  turns,  the  radius  TP  will  be  also  increased  and  diminshed  by  turns ; 
and  the  periodical  time  will  be  increased  and  diminished  in  a  ratio  com 
pounded  of  the  sesquiplicate  ratio  of  the  radius,  and  of  the  subduplicate  oi 
that  ratio  in  which  the  centripetal  force  of  the  central  body  T  is  dimin 
ished  or  increased,  by  the  increase  or  decrease  of  the  action  of  the  distant 
body  S. 

COR.  7.  It  also  follows,  from  what  was  before  laid  down,  that  the  axis 
of  the  ellipsis  described  by  the  body  P,  or  the  line  of  the  apsides,  does  as 
to  its  angular  motion  go  forwards  and  backwards  by  turns,  but  more  for 
wards  than  backwards,  and  by  the  excess  of  its  direct  motion  is  in  the 
whole  carried  forwards.  For  the  force  with  which  the  body  P  is  urged  to 
the  body  T  at  the  quadratures,  where  the  force  MN  vanishes,  is  compound 
ed  of  the  force  LM  and  the  centripetal  force  with  which  the  body  T  at 
tracts  the  body  P.  The  first  force  LM,  if  the  distance  PT  be  increased,  is 
increased  in  nearly  the  same  proportion  with  that  distance,  and  the  other 
force  decreases  in  the  duplicate  ratio  of  the  distance ;  and  therefore  the 
sum  of  these  two  forces  decreases  in  a  less  than  the  duplicate  ratio  of  the 
distance  PT ;  and  therefore,  by  Cor.  1,  Prop.  XLV,  will  make  the  line  of 
the  apsides,  or,  which  is  the  same  thing,  the  upper  apsis,  to  go  backward. 
But  at  the  conjunction  and  opposition  the  force  with  which  the  body  P  is 
urged  towards  the  body  T  is  the  difference  of  the  force  KL,  and  of  the 
force  with  which  the  body  T  attracts  the  body  P  ;  and  that  difference,  be 
cause  the  force  KL  is  very  nearly  increased  in  the  ratio  of  the  distance 
PT,  decreases  in  more -than  the  duplicate  ratio  of  the  distance  PT ;  and 
therefore,  by'  Cor.  1,  Prop.  XLV,  causes  the  line  of  the  apsides  to  go  for 
wards.  In  the  places  between  the  syzygies  and  the  quadratures,  the  motion 


SEC.    Xl.J  OF    NATURAL    PHILOSOPHY.  207 

of  the  line  of  the  apsides  depends  upon  both  <  f  these  causes  conjuncdy,  so 
that  it  either  goes  forwards  or  backwards  in  proportion  to  the  excess  ol 
one  of  these  causes  above  the  other.  Therefore  since  the  force  KL  in  the 
syzygies  is  almost  twice  as  great  as  the  force  LM  in  the  quadratures,  the 
excess  will  be  on  the  side  of  the  force  KL,  and  by  consequence  the  line  of 
the  apsides  will  be  carried  forwards.  The  truth  of  this  arid  the  foregoing 


IE 

Corollary  will  be  more  easily  understood  by  conceiving  the  system  of  the 
two  bodies  T  and  P  to  be  surrounded  on  every  side  by  several  bodies  S, 
S,  S,  dec.,  disposed  about  the  orbit  ESE.  For  by  the  actions  of  these  bo 
dies  the  action  of  the  body  T  will  be  diminished  on  every  side,  and  decrease 
in  more  than  a  duplicate  ratio  of  the  distance. 

COR.  8.  IJut  since  the  progress  or  regress  of  the  apsides  depends  upon 
the  decrease  of  the  centripetal  force,  that  is,  upon  its  being  in  a  greater  or 
less  ratio  than  the  duplicate  ratio  of  the  distance  TP,  in  the  passage  of 
the  body  from  the  lower  apsis  to  the  upper ;  and  upon  a  like  increase  in 
its  return  to  the  lower  apsis  again  ;  and  therefore  becomes  greatest  where 
the  proportion  of  the  force  at  the  upper  apsis  to  the  force  at  the  lower  ap 
sis  recedes  farthest  from  the  duplicate  ratio  of  the  distances  inversely ;  it 
is  plain,  that,  when  the  apsides  are  in  the  syzygies,  they  will,  by  reason  of 
the  subducting  force  KL  or  NM  —  LM,  go  forward  more  swiftly  ;  and  in 
the  quadratures  by  the  additional  force  LM  go  backward  more  slowly. 
Because  the  velocity  of  the  progress  or  slowness  of  the  regress  is  continued 
for  a  long  time ;  this  inequality  becomes  exceedingly  great. 

COR.  9.  If  a  body  is  obliged,  by  a  force  reciprocally  proportional  to  the 
square  of  its  distance  from  any  centre,  to  revolve  in  an  ellipsis  round  that 
centre ;  and  afterwards  in  its  descent  from  the  upper  apsis  to  the  lower 
apsis,  that  force  by  a  perpetual  accession  of  new  force  is  increased  in  more 
than  a  duplicate  ratio  of  the  diminished  distance ;  it  is  manifest  that  the 
body,  being  impelled  always  towards  the  centre  by  the  perpetual  accession 
of  this  new  force,  will  incline  more  towards  that  centre  than  if  it  were 
urged  by  that  force  alone  which  decreases  in  a  duplicate  ratio  of  the  di 
minished  distance,  and  therefore  will  describe  an  orbit  interior  to  that 
elliptical  orbit,  and  at  the  lower  apsis  approaching  nearer  to  the  centre 
than  before.  Therefore  the  orbit  by  the  accession  of  this  new  force  will 
become  more  eccentrical.  If  now,  while  the  body  is  returning  from  the 
lower  to  the  upper  apsis,  it  should  decrease  by  the  same  degrees  by  which 
it  increases  before  the  body  would  return  to  its  first  distance;  and  there- 


THE    MATHEMATICAL    PRINCIPLES  [BOOK    I. 

fore  if  the  force  decreases  in  a  yet  greater  ratio,  the  body,  being  now  less 
attracted  than  before,  will  ascend  to  a  still  greater  distance,  and  so  the  ec 
centricity  of  the  orbit  will  be  increased  still  more.  Therefore  if  the  ratio 
of  the  increase  and  decrease  of  the  centripetal  force  be  augmented  each 
revolution,  the  eccentricity  will  be  augmented  also ;  and,  on  the  contrary, 
if  that  ratio  decrease,  it  will  be  diminished. 

Now,  therefore,  in  the  system  of  the  bodies  T,  P,  S,  when  the  apsides  of 
the  orbit  FAB  are  in  the  quadratures,  the  ratio  of  that  increase  and  de 
crease  is  least  of  all,  and  becomes  greatest  when  the  apsides  are  in  the 
syzygies.  If  the  apsides  are  placed  in  the  quadratures,  the  ratio  near  the 
apsides  is  less,  and  near  the  syzygies  greater,  than  the  duplicate  ratio  of  the 
distances  :  and  from  that  Greater  ratio  arises  a  direct  motion  of  the  line  of 

7  o 

the  apsides,  as  was  just  now  said.  But  if  we  consider  the  ratio  of  the 
whole  increase  or  decrease  in  the  progress  between  the  apsides,  this  is  less 
than  the  duplicate  ratio  of  the  distances.  The  force  in  the  lower  is  to  the 
force  in  the  upper  apsis  in  less  than  a  duplicate  ratio  of  the  distance  of  the 
upper  apsis  from  the  focus  of  the  ellipsis  to  the  distance  of  the  lower  apsis 
from  the  same  focus  ;  and,  contrariwise,  when  the  apsides  are  placed  in  the 
syzygies,  the  force  in  the  lower  apsis  is  to  the  force  in  the  upper  apsis  in  a 
greater  than  a  duplicate  ratio  of  the  distances.  For  the  forces  LM  in  the 
quadratures  added  to  the  forces  of  the  body  T  compose  forces  in  a  less  ra 
tio  ;  and  the  forces  KL  in  the  syzygies  subducted  from  the  forces  of  the 
body  T,  leave  the  forces  in  a  greater  ratio.  Therefore  the  ratio  of  the 
whole  increase  and  decrease  in  the  passage  between  the  apsides  is  least  at 
the  quadratures  and  greatest  at  the  syzygies ;  and  therefore  in  the  passage 
of  the  apsides  from  the  quadratures  to  the  syzygies  it  is  continually  aug 
mented,  and  increases  the  eccentricity  of  the  ellipsis ;  and  in  the  passage 
from  the  syzygies  to  the  quadratures  it  is  perpetually  decreasing,  and  di 
minishes  the  eccentricity. 

COR.  10.  That  we  may  give  an  account  of  the  errors  as  to  latitude,  let 
us  suppose  the  plane  of  the  orbit  EST  to  remain  immovable;  and  from 
the  cause  of  the  errors  above  explained,  it  is  manifest,  that,  of  the  two 
forces  NM,  ML,  which  are  the  only  and  entire  cause  of  them,  the  force 
ML  acting  always  in  the  plane  of  the  orbit  PAB  never  disturbs  the  mo 
tions  as  to  latitude  ;  and  that  the  force  NM,  when  the  nodes  are  in  the 
gyzygies,  acting  also  in  the  same  plane  of  the  orbit,  does  not  at  that  time 
affect  those  motions.  But  when  the  nodes  are  in  the  quadratures,  it  dis 
turbs  tliem  very  much,  and,  attracting  the  body  P  perpetually  out  of  the 
plane  of  its  orbit,  it  diminishes  the  inclination  of  the  plane  in  the  passage 
of  the  body  from  the  quadratures  to  the  syzygies,  and  again  increases  the 
same  in  the  passage  from  the  syzygies  to  the  quadratures.  Hence  it 
comes  to  pass  that  when  the  body  is  in  the  syzygies,  the  inclination  is 
then  least  of  all,  and  returns  to  the  first  magnitude  nearly,  when  the  body 


SEC.  XL]  OF  NATURAL  PHILOSOPHY.  209 

arrives  at  the  next  node.     But  if  the  nodes  are  situate  at  the  octants  after 
the  quadratures,  that  is,  between  C  and  A,  D  and  B,  it  will  appear,  from 

ii  C  L 


E 

wnat  was  just  now  shewn,  that  in  the  passage  of  the  body  P  from  either 
node  to  the  ninetieth  degree  from  thence,  the  inclination  of  the  plane  is 
perpetually  diminished ;  then,  in  the  passage  through  the  next  45  degrees 
to  the  next  quadrature,  the  inclination  is  increased ;  and  afterwards,  again, 
in  its  passage  through  another  45  degrees  to  the  next  node,  it  is  dimin 
ished.  Therefore  the  inclination  is  more  diminished  than  increased,  and 
is  therefore  always  less  in  the  subsequent  node  than  in  the  preceding  one. 
And,  by  a  like  reasoning,  the  inclination  is  more  increased  than  diminish 
ed  when  the  nodes  are  in  the  other  octants  between  A  and  D,  B  and  C. 
The  inclination,  therefore,  is  the  greatest  of  all  when  the  nodes  are  in  the 
syzygies  In  their  passage  from  the  syzygies  to  the  quadratures  the  incli 
nation  is  diminished  at  each  appulse  of  the  body  to  the  nodes  :  and  be 
comes  least  of  all  when  the  nodes  are  in  the  quadratures,  and  the  body  in 
the  syzygies  ;  then  it  increases  by  the  same  degrees  by  which  it  decreased 
before ;  and,  when  the  nodes  come  to  the  next  syzygies,  returns  to  its 
former  magnitude. 

COR.  11.  Because  when  the  nodes  are  in  the  quadratures  the  body  P  is 
perpetually  attracted  from  the  plane  of  its  orbit ;  and  because  this  attrac 
tion  is  made  towards  S  in  its  passage  from  the  node  C  through  the  con 
junction  A  to  the  node  D  ;  and  to  the  contrary  part  in  its  passage  from  the 
node  D  through  the  opposition  B  to  the  node  C;  it  is  manifest  that,  in  its 
motion  from  the  node  C,  the  body  recedes  continually  from  the  former 
plane  CD  of  its  orbit  till  it  comes  to  the  next  node;  and  therefore  at  that 
node,  being  now  at  its  greatest  distance  from  the  first  plane  CD,  it  will 
pass  through  the  plane  of  the  orbit  EST  not  in  D,  the  other  node  of  that 
plane,  but  in  a  point  that  lies  nearer  to  the  body  S,  which  therefore  be 
comes  a  new  place  of  the  node  in,  antecedent ia  to  its  former  place.  And, 
by  a  like  reasoning,  the  nodes  will  continue  to  recede  in  their  passage 
from  this  node  to  the  next.  The  nodes,  therefore,  when  situate  in  the 
quadratures,  recede  perpetually ;  and  at  the  syzygies,  where  no  perturba 
tion  can  be  produced  in  the  motion  as  to  latitude,  are  quiescent :  in  the  in 
termediate  places  they  partake  of  both  conditions,  and  recede  more  slowly  ; 
and,  therefore,  being  always  either  retrograde  or  stationary,  they  will  be 
carried  backwards,  or  in  atitecedentia,  each  revolution. 

COR.  12.  All  the  errors  described  in  these  corrollaries  arc  a  little  greater 

14 


210  THE    MATHEMATICAL    PRINCIPLES  BOOK    L 

at  the  conjunction  of  the  bodies  P,  S,  than  at  their  opposition ;  because 
the  generating  forces  NM  and  ML  are  greater. 

COR.  13.  And  since  the  causes  and  proportions  of  the  errors  and  varia 
tions  mentioned  in  these  Corollaries  do  not  depend  upon  the  magnitude  of 
the  body  S,  it  follows  that  all  things  before  demonstrated  will  happen,  if 
the  magnitude  of  the  body  S  be  imagined  so  great  as  that  the  system  of  the 
two  bodies  P  and  T  may  revolve  about  it.  And  from  this  increase  of  the 
body  S,  and  the  consequent  increase  of  its  centripetal  force,  from  which  the 
errors  of  the  body  P  arise,  it  will  follow  that  all  these  errors,  at  equal  dis 
tances,  will  be  greater  in  this  case,  than  in  the  other  where  the  body  S  re 
volves  about  the  system  of  the  bodies  P  and  T. 

COR.  14.  But  since  the  forces  NM,  ML,  when  the  body  S  is  exceedingly 
distant,  are  very  nearly  as  the  force  SK  and  the  ratio  PT  to  ST  con- 
junctly  ;  that  is,  if  both  the  distance  PT,  and  the  absolute  force  of  the  body 
8  be  given,  as  ST3  reciprocally  :  and  since  those  forces  NM,  ML  are  the 
causes  of  all  the  errors  and  effects  treated  of  in  the  foregoing  Corollaries; 
it  is  manifest  that  all  those  effects,  if  the  system  of  bodies  T  and  P  con 
tinue  as  before,  and  only  the  distance  ST  and  the  absolute  force  of  the  body 
S  be  changed,  will  be  very  nearly  in  a  ratio  compounded  of  the  direct  ratio 
of  the  absolute  force  of  the  body  S,  and  the  triplicate  inverse  ratio  of  the 
distance  ST.  Hence  if  the  system  of  bodies  T  and  P  revolve  about  a  dis 
tant  body  S,  those  forces  NM,  ML,  and  their  eifl  ts,  will  be  (by  Cor.  2  and 
6,  Prop  IV)  reciprocally  in  a  duplicate  ratio  c/f  the  periodical  time.  And 
thence,  also,  if  the  magnitude  of  the  bodv  S  be  proportional  to  its  absolute 
force,  those  forces  NM,  ML,  and  their  effects,  will  be  directly  as  the  cube 
of  the  apparent  diameter  of  the  distant  body  S  viewed  from  T,  and  so  vice 
versa.  For  these  ratios  are  the  same  as  the  compounded  ratio  above  men 
tioned. 

COR.  15.  And  because  if  the  orbits  ESE  and  PAB,  retaining  their  fig 
ure,  proportions,  and  inclination  to  each  other,  should  alter  their  magni 
tude ;  arid  the  forces  of  the  bodies  S  and  T  should  either  remain,  or  be 
changed  in  any  given  ratio ;  these  forces  (that  is,  the  force  of  the  body  T, 
which  obliges  the  body  P  to  deflect  from  a  rectilinear  course  into  the  orbit 
PAB,  and  the  force  of  the  body  S,  which  causes  the  body  P  to  deviate  from 
that  orbit)  would  act  always  in  the  same  manner,  and  in  the  same  propor 
tion  :  it  follows,  that  all  the  effects  will  be  similar  and  proportional,  arid 
the  times  of  those  effects  proportional  also  ;  that  is,  that  all  the  linear  er 
rors  will  be  as  tne  diameters  of  the  orbits,  the  angular  errors  the  same  as 
before ;  and  the  times  of  similar  linear  errors,  or  equal  angular  errors?  as 
the  periodical  times  of  the  orbits. 

COR.  16.  Therefore  if  the  figures  of  the  orbits  and  their  inclination  to 
each  other  be  given,  and  the  magnitudes,  forces,  arid  distances  of  the  bodies 
he  any  how  changed,  we  may.  from  the  errors  and  times  of  those  errors  in 


SEC.    XI.]  OF    NATURAL    PHILOSOPHY.  2 \\ 

one  case,  collect  very  nearly  the  errors  and  times  of  the  errors  in  any  other 
case.  But  this  may  be  done  more  expeditiously  by  the  following  method. 
The  forces  NM;  ML,  other  things  remaining  unaltered,  are  as  the  radius 
TP  ;  and  their  periodical  effects  (by  Cor.  2,  Lein.  X)  are  as  the  forces  and 
the  square  of  the  periodical  time  of  the  body  P  conjunctly.  These  are  the 
linear  errors  of  the  body  P  ;  and  hence  the  angular  errors  as  they  appear 
from  the  centre  T  (that  is,  the  motion  of  the  apsides  and  of  the  nodes,  and  all 
the  apparent  errors  as  to  longitude  and  latitude)  are  in  each  revolution  of 
the  body  P  as  the  square  of  the  time  of  the  revolution,  very  nearly.  Let 
these  ratios  be  compounded  with  the  ratios  in  Cor.  14,  and  in  any  system 
of  bodies  T,  P,  S,  where  P  revolves  about  T  very  near  to  it,  and  T  re 
volves  about  S  at  a  great  distance,  the  angular  errors  of  the  body  P,  ob 
served  from  the  centre  T,  will  be  in  each  revolution  of  the  body  P  as  the 
square  of  the  periodical  time  of  the  body  P  directly,  and  the  square  of  the 
periodical  time  of  the  body  T  inversely.  And  therefore  the  mean  motion 
of  the  line  of  the  apsides  will  be  in  a  given  ratio  to  the  mean  motion  of 
the  nodes ;  and  both  those  motions  will  be  as  the  periodical  time  of  the 
body  P  directly,  and  the  square  of  the  periodical  time  of  the  body  T  in 
versely.  The  increase  or  diminution  of  the  eccentricity  and  inclination  of 
the  orbit  PAB  makes  no  sensible  variation  in  the  motions  of  the  apsides* 
and  nodes,  unless  that  inc/case  or  diminution  be  very  great  indeed. 

COR.  17.  Sines  the  line  LM  becomes  sometimes  greater  and  sometimes 
less  than  the  radius  PT,  let  the  mean  quantity  of  the  force  LM  be  expressed 
E  C 


sa— - -•::-.•.•::::::; 


by  that  radius  PT ;  and  then  that  mean  force  will  be  to  the  mean  force 
SK  or  SN  (which  may  be  also  expressed  by  ST)  as  the  length  PT  to  the 
length  ST.  But  the  mean  force  SN  or  ST,  by  which  the  body  T  is  re 
tained  in  the  orbit  it  describes  about  S,  is  to  the  force  with  which  the  body  P 
is  retained  in  its  orbit  about  T  in  a  ratio  compounded  of  the  ratio  of  the 
radius  ST  to  the  radius  PT,  and  the  duplicate  ratio  of  the  periodical  time 
of  the  body  P  about  T  to  the  periodical  time  of  the  body  T  about  S.  And, 
ex  cequo,  the  mean  force  LM  is  to  the  force  by  which  the  body  P  is  retain 
ed  in  its  orbit  about  T  (or  by  which  the  same  body  P  might  revolve  at  the 
distance  PT  in  the  same  periodical  time  about  any  immovable  point  T)  in 
the  same  duplicate  ratio  of  the  periodical  times.  The  periodical  times 
therefore  being  given,  together  with  the  distance  PT,  the  mean  force  LM 
is  also  given  ;  and  that  force  being  given,  there  is  given  also  the  force  MN, 
very  nearly,  by  the  analogy  of  the  lines  PT  and  MN. 


212  THE    MATHEMATICAL    PRINCIPLES  [BoOK    I 

Con.  IS.  By  tlie  same  laws  by  which  the  body  P  revolves  about  the 
body  T,  let  us  suppose  many  fluid  bodies  to  move  round  T  at  equal  dis 
tances  from  it ;  and  to  be  so  numerous,  that  they  may  all  become  contiguous 
to  each  other,  so  as  to  form  a  fluid  annul  us,  or  ring,  of  a  round  figure,  and 
concentrical  to  the  body  T;  and  the  several  parts  of  this  annulus,  perform 
ing  their  motions  by  the  same  law  as  the  body  P,  will  draw  nearer  to  the 
body  T,  and  move  swifter  in  the  conjunction  and  opposition  of  themselves 
and  the  body  S,  than  in  the  quadratures.  And  the  nodes  of  this  annulus, 
or  its  intersections  with  the  plane  of  the  orbit  of  the  body  S  or  T,  will  rest 
at  the  syzygies  ;  but  out  of  the  syzygies  they  will  be  carried  backward,  or 
in.  antecedentia  ;  with  the  greatest  swiftness  in  the  quadratures,  and  more 
slowly  in  other  places.  The  inclination  of  this  annulus  also  will  vary,  and 
its  axis  will  oscillate  each  revolution,  and  when  the  revolution  is  completed 
will  return  to  its  former  situation,  except  only  that  it  will  be  carried  round 
a  little  by  the  precession  of  the  nodes. 

COR.  19.  Suppose  now  the  spherical  body  T,  consisting  of  some  matter 
not  fluid,  to  be  enlarged,  and  to  extend  its  'If  on  every  side  as  far  as  that 
annulus,  and  that  a  channel  were  cut  all  round  its  circumference  contain 
ing  water  j  and  that  this  sphere  revolves  uniformly  about  its  own  axis  in 
the  same  periodical  time.  This  water  being  accelerated  and  retarded  by 
turns  (as  in  the  last  Corollary),  will  be  swifter  at  the  syzygies,  and  slower 
at  the  quadratures,  than  the  surface  of  the  globe,  and  so  will  ebb  and  flow  in 
its  channel  after  the  manner  of  the  sea.  If  the  attraction  of  the  body  S  were 
taken  away,  the  water  would  acquire  no  motion  of  flux  and  reflux  by  revolv- 
.ng  round  the  quiescent  centre  of  the  globe.  The  case  is  the  same  of  a  globe 
moving  uniformly  forwards  in  a  right  line,  and  in  the  mean  time  revolving 
about  its  centre  (by  Cor.  5  of  the  Laws  of  Motion),  and  of  a  globe  uni 
formly  attracted  from  its  rectilinear  course  (by  Cor.  6,  of  the  same  Laws). 
But  let  the  body  S  come  to  act  upon  it,  and  by  its  unequable  attraction  the 
A\ater  will  receive  this  new  motion  ;  for  there  will  be  a  stronger  attraction 
upon  that  part  of  the  water  that  is  nearest  to  the  body,  and  a  weaker  upon 
that  part  which  is  more  remote.  And  the  force  LM  will  attract  the  water 
downwards  at  the  quadratures,  and  depress  it  as  far  as  the  syzygies  ;  and  the 
force  KL  will  attract  it  upwards  in  the  syzygies,  and  withhold  its  descent, 
and  make  it  rise  as  far  as  the  quadratures ;  except  only  in  so  far  as  the 
motion  of  flux  and  reflux  may  be  directed  by  the  channel  of  the  water,  and 
be  a  little  retarded  by  friction. 

COR.  20.  If,  now,  the  annulus  becomes  hard,  and  the  globe  is  diminished, 
the  motion  of  flux  and  reflux  will  cease  ;  but  the  oscillating  motion  of  the 
inclination  and  the  praecession  of  the  nodes  will  remain.  Let  the  globe 
have  the  same  axis  with  the  annulus,  and  perform  its  revolutions  in  the 
same  times,  and  at  its  surface  touch  the  annulus  within,  and  adhere  to  it; 
then  the  globe  partaking  of  the  motion  of  the  annulus,  this  whole  compares 


SEC.    XI.  OF    NATURAL    PHILOSOPHY.  213 

will  oscillate,  and  the  nodes  will  go  backward,  for  the  globe,  as  \ve  shall 
shew  presently,  is  perfectly  indifferent  to  the  receiving  of  all  impressions. 
The  greatest  angle  of  the  inclination  of  the  annulus  single  is  when  the 
nodes  are  in  the  syzygies.  Thence  in  the  progress  of  the  nodes  to  the 
quadratures,  it  endeavours  to  diminish  its  inclination,  and  by  that  endea 
vour  impresses  a  motion  upon  the  whole  globe.  The  globe  retains  this 
motion  impressed,  till  the  annulus  by  a  contrary  endeavour  destroys  that 
motion,  and  impresses  a  new  motion  in  a  contrary  direction.  And  by  this 
means  the  greatest  motion  of  the  decreasing  inclination  happens  when  the 
nodes  are  in  the  quadratures;  and  the  least  angle  of  inclination  in  the  octants 


B 

after  the  quadratures ;  and,  again,  the  greatest  motion  of  roclination  happens 
when  the  nodes  are  in  the  syzygies ;  and  the  greatest  angle  of  reclination  in 
the  octants  following.  And  the  case  is  the  same  of  a  globe  without  this  an 
nulus,  if  it  be  a  little  higher  or  a  little  denser  in  the  equatorial  than  in  the 
polar  regions  :  for  the  excess  of  that  matter  in  the  regions  near  the  equator 
supplies  the  place  of  the  annulus.  And  though  we  should  suppose  the  cen 
tripetal  force  of  this  globe  to  be  any  how  increased,  so  that  all  its  parts 
were  to  tend  downwards,  as  the  parts  of  our  earth  gravitate  to  the  centre, 
yet  the  phenomena  of  this  and  the  preceding  Corollary  would  scarce  be  al 
tered  ;  except  that  the  places  of  the  greatest  and  least  height  of  the  water 
will  be  different :  for  the  water  is  now  no  longer  sustained  and  kept  in  its 
orbit  by  its  centrifugal  force,  but  by  the  channel  in  which  it  flows.  And, 
besides,  the  force  LM  attracts  the  water  downwards  most  in  the  quadra 
tures,  and  the  force  KL  or  NM  —  LM  attracts  it  upwards  most  in  the 
syzygies.  And  these  forces  conjoined  cease  to  attract  the  water  downwards, 
and  begin  to  attract  it  upwards  in  the  octants  before  the  syzygies  ;  and 
cease  to  attract  the  water  upwards,  and  begin  to  attract  the  water  down 
wards  in  the  octants  after  the  syzygies.  And  thence  the  greatest  height  of 
the  water  may  happen  about  the  octants  after  the  syzygies ;  and  the  least 
height  about  the  octants  after  the  quadratures ;  excepting  only  so  far  as  the 
motion  of  ascent  or  descent  impressed  by  these  forces  may  by  the  vis  insita 
of  the  water  continue  a  little  longer,  or  be  stopped  a  little  sooner  by  impe 
diments  in  its  channel. 

COR.  21.  For  the  same  reason  that  redundant  matter  in  the  equatorial 
regions  of  a  globe  causes  the  nodes  to  go  backwards,  and  therefore  by  the 
increase  of  that  matter  that  retrogradation  is  increased,  by  the  diminution 
is  diminished,  and  by  the  removal  quite  ceases :  it  follows,  that,  if  more  than 


214  THE    MATHEMATICAL    PRINCIPLES  [BOOK    I 

that  redundant  matter  be  taken  away,  that  is,  if  the  globe  be  either  more 
depressed,  or  of  a  more  rare  consistence  near  the  equator  than  near  the 
poles,  there  will  arise  a  motion  of  the  nodes  in  consequentia. 

COR.  22.  And  thence  from  the  motion  of  the  nodes  is  known  the  consti 
tution  of  the  globe.  That  is,  if  the  globe  retains  unalterably  the  same  poles, 
and  the  motion  (of  the  nodes)  be  in.  antecedetitia,  there  is  a  redundance  oi 
the  matter  near  the  equator;  but  if  in  conseqnentia,  a  deficiency.  Sup 
pose  a  uniform  and  exactly  spherical  globe  to  be  first  at  rest  in  a  free  space  : 
then  by  some  impulse  made  obliquely  upon  its  superficies  to  be  driven  from 
its  place,  and  to  receive  a  motion  partly  circular  and  partly  right  forward. 
Because  this  globe  is  perfectly  indifferent  to  all  the  axes  that  pass  through 
its  centre,  nor  has  a  greater  propensity  to  one  axis  or  to  one  situation  oi 
the  axis  than  to  any  other,  it  is  manifest  that  by  its  own  force  it  will  never 
change  its  axis,  or  the  inclination  of  it.  Let  now  this  globe  be  impelled 
obliquely  by  a  new  impulse  in  the  same  part  of  its  superficies  as  before . 
and  since  the  effect  of  an  impulse  is  not  at  all  changed  by  its  coming  sooner 
or  later,  it  is  manifest  that  these  two  impulses,  successively  impressed,  will 
produce  the  same  motion  as  if  they  were  impressed  at  the  same  time :  that 
is,  the  same  motion  as  if  the  globe  had  been  impelled  by  a  simple  force 
compounded  of  them  both  (by  Cor.  2,  of  the  Laws),  that  is,  a  simple  motion 
about  an  axis  of  a  given  inclination.  And  the  case  is  the  same  if  the  sec 
ond  impulse  were  made  upon  any  other  place  of  the  equator  of  the  first 
motion ;  and  also  if  the  first  impulse  were  made  upon  any  place  in  the 
equator  of  the  motion  which  would  be  generated  by  the  second  impulse 
alone;  and  therefore,  also,  when  both  impulses  are  made  in  any  places 
whatsoever ;  for  these  impulses  will  generate  the  same  circular  motion  as 
if  they  were  impressed  together,  and  at  once,  in  the  place  of  the  intersec 
tions  of  the  equators  of  those  motions,  which  would  be  generated  by  each 
of  them  separately.  Therefore,  a  homogeneous  and  perfect  globe  will  not 
retain  several  distinct  motions,  but  will  unite  all  those  that  are  impressed 
on  it,  and  reduce  them  into  one;  revolving, as  far  as  in  it  lies,  always  with 
a  simple  and  uniform  motion  about  one  single  given  axis,  with  an  inclina 
tion  perpetually  invariable.  And  the  inclination  of  the  axis,  or  the  velocity 
of  the  rotation,  will  not  be  changed  by  centripetal  force.  For  if  the  globe 
be  supposed  to  be  divided  into  two  hemispheres,  by  any  plane  whatsoever 
passing  through  its  own  centre,  and  the  centre  to  which  the  force  is  direct 
ed,  that  force  will  always  urge  each  hemisphere  equally  ;  and  therefore  will 
not  incline  the  globe  any  way  as  to  its  motion  round  its  own  axis.  But 
let  there  be  added  any  where  between  the  pole  and  the  equator  a  heap  oi 
new  matter  like  a  mountain,  and  this,  by  its  perpetual  endeavour  to  recede 
from  the  centre  of  its  motion,  will  disturb  the  motion  of  the  globe,  and 
cause  its  poles  to  wander  about  its  superficies,  describing  circles  about 
themselves  and  their  opposite  points.  Neither  can  this  enormous  evagatior 


XL]  OF    NATURAL    PHILOSOPHY.  2 In 

of  the  poles  be  corrected,  unless  by  placing  that  mountain  ei  '.  er  in  one  ol 
the  poles;  in  which  case,  by  Cor.  21,  the  nodes  of  the  equator  will  go  for 
wards  ;  or  in  the  equatorial  regions,  in  which  case,  by  Cor.  20,  the  nodes 
will  go  backwards:  or,  lastly,  by  adding  on  the  other  side  of  the  axis  anew 
quantity  of  matter,  by  which  the  mountain  may  be  balanced  in  its  motion; 
and  then  the  nodes  will  either  go  forwards  or  backwards,  as  the  mountain 
and  this  newly  added  matter  happen  to  be  nearer  to  the  pole  or  to  the 
equator. 

PROPOSITION  LXV1I.     THEOREM  XXVII. 

The  same  laics  of  attraction  being'  supposed,  I  say,  that  the  exterior  body 
S  does,  by  radii  dra.cn  to  the  point  O,  the  common  centre  of  gravity 
of  the  interior  bodies  P  and  T,  describe  round  that  centre  areas  more 
proportional  to  the  times,  and  an  orbit  more  approaching  to  the  form 
of  an  ellipsis  having  its  focus  in  that  cen  > •.-.  than,  it  can  describe 
round  the  innermost  and  greatest  body  T  by  ra  Hi  drawn  to  that 
body. 
For  the  attractions  of  the  body  S  towards  T  and 

P  compose  its  absolute  attraction,  which  is  more 

directed  towards  O,  the  common  centre  of  gravity  S(i 

of  the  bodies  T  and  P,  than  it  is  to   the  «.  reatest 

body  T  ;  and  which  is  more  in  a  reciprocal  propor 
tion  to  the  square  of  the  distance  SO,  than  it  is  to  the  square  of  the  distance 

ST  :  as  will  easily  appear  by  a  little  consideration. 

PROPOSITION  LXVIII.    THEOREM  XXVIII. 

The  same  laws  of  attraction  supposed,  I  say,  that  the  exterior  body  S 
will,  by  radii  drawn  to  O,  the  common  centre  of  gravity  of  the  interior 
bodies  P   and    T,    describe    round  that    centre    areas  more  propor 
tional  to  the  times,  and  an  orbit  more  approaching  to  the  form  of  an 
ellipsis  having  its  focus  in  that  centre,  if  the  innermost  and  greatest 
body  be  agitated  by  these  attractions  as  well  as  the  rest,  than  it  would 
do  if  that  body  were  either  at  rest  as  not  attracted,  or  were  much  tnore 
or  much  less  attracted,  or  much  more  or  much  less  agitated. 
This  may  be  demonstrated  after  the  same  manner  as  Prop.  LXVI,  but 
by  a  more  prolix  reasoning,  which  I  therefore  pass  over.     It  will  be  suf 
ficient  to  consider  it  after  this  manner.     From  the  demonstration  of  the 
last  Proposition  it  is  plain,  that  the  centre,  towards  which  the  body  S  is 
urged  by  the  two  forces  conjunctly,  is  very  near  to  the  common  centre  of 
gravity  of  those  two  other  bodies.     If  this  centre  were  to  coincide  with  that 
common  centre,  and  moreover  the  common  centre  of  gravity  of  all  the  three 
bodies  were  at  rest,  the  body  S  on  one  side,  and  the  common   centre  of 
gravity  of  the  other  two  bodies  on  the  other  side,  would  describe  true  ellip* 


216  THE    MATHEMATICAL    PRINCIPLES  -  [BOOK    1 

ses  about  that  quiescent  common  centre.  This  appears  from  Cor.  2,  Pro]) 
LVIII,  compared  with  what  was  demonstrated  in  Prop.  LX1V,  and  LXY 
Now  this  accurate  elliptical  motion  will  be  disturbed  a  little  by  the  dis 
tance  of  the  centre  of  the  two  bodies  from  the  centre  towards  which  tht 
third  body  S  is  attracted.  Let  there  be  added,  moreover,  a  motion  to  the 
Bommon  centre  of  the  three,  and  the  perturbation  will  be  increased  yet 
more.  Therefore  the  perturbation  is  least  when  the 
common  centre  of  the  three  bodies  is  at  rest;  that 
I  is,  when  the  innermost  and  greatest  body  T  is  at 
tracted  according  to  the  same  law  as  the  rest  are ; 
and  is  always  greatest  when  the  common  centre  of 
the  three,  by  the  diminution  of  the  motion  of  the  body  T,  begins  to  be 
moved,  and  is  more  and  more  agitated. 

COR.  And  hence  if  more  lesser  bodies  revolve  about  the  great  one,  it 
may  easily  be  inferred  that  the  orbits  described  will  approach  nearer  to 
ellipses ;  and  the  descriptions  of  areas  will  be  more  nearly  equable,  if  all 
the  bodies  mutually  attract  and  agitate  each  other  with  accelerative  forces 
that  are  as  their  absolute  forces  directly,  and  the  squares  of  the  distances 
inversely  :  and  if  the  focus  of  each  orbit  be  placed  in  the  common  centre 
of  gravity  of  all  the  interior  bodies  (that  is.  if  the  focus  of  the  first  and  in 
nermost  orbit  be  placed  in  the  centre  of  gravity  of  the  greatest  and  inner 
most  body  :  the  focus  of  the  second  orbit  in  the  common  centre  of  gravity 
of  the  two  innermost  bodies;  the  focus  of  the  third  orbit  in  the  common 
centre  of  gravity  of  the  three  innermost ;  and  so  on),  than  if  the  innermost 
body  were  at  rest,  and  was  made  the  common  focus  of  all  the  orbits. 

PROPOSITION  LXIX.     THEOREM  XXIX. 

fn  a  system  of  several  bodies  A,  B,  C,  D,  $*c.,  if  any  one  of  those  bodies, 
as  A,  attract  all  the  rest,  B,  C,  D,  $*c.,with  accelerative  forces  that  are 
reciprocally  as  the  squares  of  the  distances  from  the  attracting  body  ; 
and  another  body,  as  B,  attracts  also  the  rest.  A,  C,  D,  $-c.,  with  forces 
that  are  reciprocally  as  the  squares  of  the  distances  from  the  attract 
ing  body  ;  the  absolute  forces  of  the  attracting  bodies  A  and  B  will 
be  to  each  other  as  those  very  bodies  A  and  B  to  which  those  forces 
belong. 

For  the  accelerative  attractions  of  all  the  bodies  B,  C,  D,  towards  A, 
are  by  the  supposition  equal  to  each  other  at  equal  distances ;  and  in  like 
manner  the  accelerative  attractions  of  all  the  bodies  towards  B  are  also 
equal  to  each  other  at  equal  distances.  But  the  absolute  attractive  force 
of  the  body  A  is  to  the  absolute  attractive  force  of  the  body  B  as  the  ac- 
eelerative  attraction  of  all  the  bodies  towards  A  to  the  accelerative  attrac 
tion  of  all  the  bodies  towards  B  at  equal  distances ;  and  so  is  also  the  ac 
celerative  attraction  of  the  body  B  to*vards  A  to  the  accelerative  attraction 


SEC.    XI]  OF    NATURAL    PHILOSOPHY.  21  T 

of  the  body  A  towards  B.  But  the  accelerative  attraction  of  the  body  B 
towards  A  is  to  the  accelerative  attraction  of  the  body  A  towards  B  as  the 
mass  of  the  body  A  to  the  mass  of  the  body  B  ;  because  the  motive  forces 
which  (by  the  2d,  7th,  and  8th  Definition)  are  as  the  accelerative  forces 
and  the  bodies  attracted  conjunctly  are  here  equal  to  one  another  by  the 
third  Law.  Therefore  the  absolute  attractive  force  of  the  body  A  is  to  the 
absolute  attractive  force  of  the  body  B  aa  the  mass  of  the  body  A  to  the 
mass  of  the  body  B.  Q.E.D. 

COR.  1.  Therefore  if  each  of  the  bodies  of  the  system  A,  B,  C,  D,  &c. 
does  singly  attract  all  the  rest  with  accelerative  forces  that  are  reciprocally 
as  the  squares  of  the  distances  from  the  attracting  body,  the  absolute  forces 
of  all  those  bodies  will  be  to  each  other  as  the  bodies  themselves. 

COR.  2.  By  a  like  reasoning,  if  each  of  the  bodies  of  the  system  A,  B, 
C,  D,  &c.,  do  singly  attract  all  the  rest  with  accelerative  forces,  which  are 
either  reciprocally  or  directly  in  the  ratio  of  any  power  whatever  of  the 
distances  from  the  attracting  body :  or  which  are  defined  by  the  distances 
from  each  of  the  attracting  bodies  according  to  any  common  law  :  it  is  plain 
that  the  absolute  forces  of  those  bodies  are  as  the  bodies  themselves. 

COR.  3.  In  a  system  of  bodies  whose  forces  decrease  in  the  duplicate  ra 
tio  of  the  distances,  if  the  lesser  revolve  about  one  very  great  one  in  ellip 
ses,  having  their  common  focus  in  the  centre  of  that  great  body,  and  of  a 
figure  exceedingly  accurate ;  and  moreover  by  radii  drawn  to  that  great 
ody  describe  areas  proportional  to  the  times  exactly  •  the  absolute  forces 
)i  those  bodies  to  each  other  will  be  either  accurately  or  very  nearly  in  the 
ratio  of  the  bodies.  And  s  >  on  the  contrary.  This  appears  from  Cor.  of 
Prop.  XLVII1, compared  with  the  first  Corollary  of  this  Prop. 

SCHOLIUM. 

These  Propositions  naturally  lead  us  to  the  analogy  there  is  between 
centripetal  forces,  and  the  central  bodies  to  which  those  forces  used  to  be 
directed ;  for  it  is  reasonable  to  suppose  that  forces  which  are  directed  to 
bodies  should  depend  upon  the  nature  and  quantity  of  those  bodies,  as  we 
see  they  do  in  magnetical  experiments.  And  when  such  cases  occur,  we 
are  to  compute  the  attractions  of  the  bodies  by  assigning  to  each  of  their 
particles  its  proper  force,  and  then  collecting  the  sum  of  them  all.  I  here 
ue*e  the  word  attraction  in  general  for  any  endeavour,  of  what  kind  soever, 
made  by  bodies  to  approach  to  each  other;  whether  that  endeavour  arise 
from  the  action  of  the  bodies  themselves,  as  tending  mutually  to  or  agita 
ting  each  other  by  spirits  emitted;  or  whether  it  arises  from  the  action 
of  the  aether  or  of  the  air,  or  of  any  medium  whatsoever*  whether  corporeal 
or  incorporeal,  any  how  impelling  bodies  placed  therein  towards  each  other. 
In  the  same  general  sense  I  use  the  word  impulse,  not  defining  in  this  trea 
tise  the  species  or  physical  qualities  of  forces,  but  investigating  the  quantities 


THE    MATHEMATICAL    PRINCIPLES  [BOOK    ). 

and  mathematical  proportions  of  them  ;  as  I  observed  before  ir  (lie  Defi 
nitions.  In  mathematics  we  are  to  investigate  the  quantities  of  forces 
with  their  proportions  consequent  upon  any  conditions  supposed  ;  then, 
when  we  enter  upon  physics,  we  compare  those  proportions  with  the  phe 
nomena  of  Nature,  that  we  may  know  what  conditions  of  those  forces  an 
swer  to  the  several  kinds  of  attractive  bodies.  And  this  preparation  being 
made,  we  argue  more  safely  concerning  the  physical  species,  causes,  and 
proportions  of  the  forces.  Let  us  see,  then,  with  what  forces  spherical 
bodies  consisting  of  particles  endued  with  attractive  powers  in  the  manner 
above  spoken  of  must  act  mutually  upon  one  another  :  and  what  kind  of 
motions  will  follow  from  thence. 


SECTION  XII. 

Of  the  attractive  forces  of  sphcerical  bodies. 

PROPOSITION  LXX.     THEOREM  XXX. 

If  to  every  point  of  a  spherical  surface  there  tend  equal  centripetal  forces 
decreasing  in,  the  duplicate  ratio  of  the  distances  from  those  points  ; 
I  say,  that  a  corpuscle  placed  within  that  superficies  will  not  be  attract 
ed  by  those  forces  any  way. 

Let  HIKL,  be  that  sphaerical  superficies,  and  P  a 
corpuscle  placed  within.  Through  P  let  there  be 
drawn  to  this  superficies  to  two  lines  HK,  IL,  inter- 
cepting  very  small  arcs  HI,  KL  ;  and  because  (by 
Cor.  3,  Lem.  VII)  the  triangles  HPI,LPK  are  alike, 
those  arcs  will  be  proportional  to  the  distances  HP 
LP ;  and  any  particles  at  HI  and  KL  of  the  spheri 
cal  superficies,  terminated  by  right  lines  passing  through  P,  will  be  in  the 
duplicate  ratio  of  those  distances.  Therefore  the  forces  of  these  particles 
exerted  upon  the  body  P  are  equal  between  themselves.  For  the  forces  are 
as  the  particles  directly,  and  the  squares  of  the  distances  inversely.  And 
these  two  ratios  compose  the  ratio  of  equality.  The  attractions  therefore, 
being  made  equally  towards  contrary  parts,  destroy  each  other.  And  by  a 
like  reasoning  all  the  attractions  through  the  whole  spherical  superficies 
are  destroyed  by  contrary  attractions.  Therefore  the  body  P  will  not  be 
any  way  impelled  by  those  attractions.  Q.E.D. 

PROPOSITION  LXXI.     THEOREM  XXXI. 

The  same  things  supposed  as  above,  I  say,  that  a  cor pu  vie  placed  with 
out  the  sph(eric»l  superficies  is  attracted  towards  the  centre  of  tht 
sphere  wiih  a  force  reciprocally  proportional  to  the  square  of  its  dis 
tance  from  that  centre. 
Let  AHKB,  ahkb,  be  two  equal  sphaerical  superficies  described  about 


SEC.    XII.J  OF    NATURAL    PHILOSOPHY. 

the  centre  S,  s  ;  their  diameters  AB,  ab  ;  and  let  P  and  p  be  two  corpus 
cles  situate  without  the  gpheres  in  those  diameters  produced.     Let   there 


be  drawn  from  the  corpuscles  the  lines  PHK,  PIL,  phk,  pil,  cutting  off 
from  the  great  circles  AHB,  ahb,  the  equal  arcs  HK,  hk,  IL;  il ;  and  to 
those  lines  let  fall  the  perpendiculars  SD,  sd,  SE,  SP,  1R,  ir  ;  of  which  let 
SD,  sd,  cut  PL,  pi,  in  F  and  f.  Let  fall  also  to  the  diameters  the  perpen 
diculars  IQ,  iq.  Let  now  the  angles  DPE,  dpe,  vanish;  and  because  DS 
and  ds,  ES  and  es  are  equal,  the  lines  PE,  PP,  and  pe,  pf,  and  the  lineolso 
I  )F,  df  may  be  taken  for  equal ;  because  their  last  ratio,  when  the  angles 
DPE,  dpe  vanish  together,  is  the  ratio  of  equality.  These  things  then 
supposed,  it  will  be,  as  PI  to  PF  so  is  RI  to  DF,  and  as  pf  to  pi  so  is  df  or 
DF  to  ri  ;  and,  ex  cequo,  as  PI  X  pf  to  PF  X  pi  so  is  R I  to  ri,  that  is 
(by  Cor.  3,  Lem  VII),  so  is  the  arc  IH  to  the  arc  ih.  Again,  PI  is  to  PS 
as  IQ.  to  SE,  and  ps  to  pi  as  se  or  SE  to  iq  ;  and,  ex  ceqno,  PI  X  ps  to 
PS  X  pi  as  IQ.  to  iq.  And  compounding  the  ratios  PI2  X  pf  X  ps  is  to 
pi2  X  PF  X  PS,  as  IH  X  IQ  to  ih  X  iq  ;  that  is,  as  the  circular  super 
ficies  which  is  described  by  the  arc  IH,  as  the  semi-circle  AKB  revolves 
about  the  diameter  AB,  is  to  the  circular  superficies  described  by  the  arc  ih 
as  the  semi-circle  akb  revolves  about  the  diameter  ab.  And  the  forces 
with  which  these  superficies  attract  the  corpuscles  P  and  p  in  the  direction 
of  lines  tending  to  those  superficies  are  by  the  hypothesis  as  the  superficies 
themselves  directly,  and  the  squares  of  the  distances  of  the  superficies  from 
those  corpuscles  inversely;  that  is,  as  pf  X  ps  to  PF  XPS.  And  these 
forces  again  are  to  the  oblique  parts  of  them  which  (by  the  resolution  of 
forces  as  in  Cor.  2,  of  the  Laws)  tend  to  the  centres  in  the  directions  of  the 
lines  PS,  JDS-,  as  PI  to  PQ,  and  pi  to  pq  ;  that  is  (because  of  the  like  trian 
gles  PIQ  and  PSF,  piq  and  psf\  as  PS  to  PF  and  ps  to  pf.  Thence  ex 
cequO)  the  attraction  of  the  corpuscle  P  towards  S  is  to  the  attraction  of 

PF  XpfXps.        pf  X  PF  X  PS    . 
the  corpusclejo  towards  5  as ~ =—  is  to — ,  that  is, 

as  ps2  to  PS2.  And,  by  a  like  reasoning,  the  forces  with  which  the  su 
perficies  described  by  the  revolution  of  the  arcs  KL,  kl  attract  those  cor 
puscles,  will  be  as  jDS2  to  PS2.  And  in  the  same  ratio  will  be  the  foroes 
of  all  the  circular  superficies  into  which  each  of  the  sphaerical  superficies 
may  be  divided  by  taking  sd  always  equal  to  SD,  and  se  equal  to  SE.  And 
therefore,  by  composition,  the  forces  of  the  entire  spherical  superficies  ex 
erted  upon  those  corpuscles  will  be  in  the  same  ratio.  Q.E.D 


220  THE    MATHEMATICAL    PRINCIPLES  [BOOK    1 

PROPOSITION  LXXIL     THEOREM  XXXII. 

If  to  the  several  points  of  a  sphere  there  tend  equal  centripetal  forces  de 
creasing  in  a  duplicate  ratio  of  the  distances  from  those  points  ;  and 
there  be  given  both  the  density  of  the  sphere  and  the  ratio  of  the  di 
ameter  of  the  sphere  to  the  distance  of  the  corpuscle  from  its  centre  ; 
I  say,  that  the  force  with  which  the  corpuscle  is  attracted  is  propor 
tional  to  the  semi-diameter  of  the  sphere. 

For  conceive  two  corpuscles  to  be  severally  attracted  by  two  spheres,  one 
by  one,  the  other  by  the  other,  and  their  distances  from  the  centres  of  the 
spheres  to  be  proportional  to  the  diameters  of  the  spheres  respectively  ,  and 
the  spheres  to  be  resolved  into  like  particles,  disposed  in  a  like  situation 
to  the  corpuscles.  Then  the  attractions  of  one  corpuscle  towards  the  sev 
eral  particles  of  one  sphere  will  be  to  the  attractions  of  the  other  towards 
as  many  analogous  particles  of  the  other  sphere  in  a  ratio  compounded  of 
the  ratio  of  the  particles  directly,  and  the  duplicate  ratio  of  the  distances 
inversely.  But  the  particles  are  as  the  spheres,  that  is,  in  a  triplicate  ra 
tio  of  the  diameters,  and  the  distances  are  as  the  diameters ;  and  the  first 
ratio  directly  with  the  last  ratio  taken  twice  inversely,  becomes  the  ratio 
of  diameter  to  diameter.  Q.E.D. 

COR.  1.  Hence  if  corpuscles  revolve  in  circles  about  spheres  composed 
of  matter  equally  attracting,  and  the  distances  from  the  centres  of  the 
spheres  be  proportional  to  their  diameters,  the  periodic  times  will  be  equal. 
COR.  2.  And,  vice  versa,  if  the  periodic  times  are  equal,  the  distances 
will  be  proportional  to  the  diameters.  These  two  Corollaries  appear  from 
Cor.  3,  Prop.  IV. 

COR.  3.  If  to  the  several  points  of  an^  two  solids  whatever,  of  like  fig- 
are  and  equal  density,  there  tend  equal  centripetal  forces  decreasing  in  a 
duplicate  ratio  of  the  distances  from  those  points,  the  forces,  with  which 
corpuscles  placed  in  a  like  situation  to  those  two  solids  will  be  attracted 
by  them,  will  be  to  each  other  as  the  diameters  of  the  solids. 

PROPOSITION  LXXIII.     THEOREM  XXXIII. 

If  to  the  several  points  of  a  given  sphere  there  tend  equal  centripetal  forces 
decreasing  in  a  duplicate  ratio  of  the  distances  from  the  points  ;  1 
say,  that  a  corpuscle  placed  within  the  sphere  is  attracted  by  a  force 
proportional  to  its  distance  from  the  centre. 

In  the  sphere  ABCD,  described  about  the  centre  S, 
let  there  be  placed  the  corpuscle  P ;  and  about  the 
same  centre  S,  with  the  interval  SP?  conceive  de- 
|B  scribed  an  interior  sphere  PEQP.  It  is  plain  (by 
Prop.  LXX)  that  the  concentric  sphaerical  superficies, 
of  which  the  difference  AEBF  of  the  spheres  is  com 
posed,  have  no  effect  at  all  upon  the  body  P,  their  at- 


SEC.    XIL]  OF    NATURAL    PHILOSOPHY.  22\ 

tractions  being  destroyed  by  contrary  attractions.  There  remains,  there 
fore;  only  the  attraction  of  the  interior  sphere  PEQ,F.  And  (by  Prop. 
LXXII)  this  is  as  the  distance  PS.  Q.E.D. 

SCHOLIUM. 

By  the  superficies  of  which  I  here  imagine  the  solids  composed,  I  do  not 
mean  superficies  purely  mathematical,  but  orbs  so  extremely  thin,  that 
their  thickness  is  as  nothing;  that  is,  the  evanescent  orbs  of  which  the  sphere 
will  at  last  consist  when  the  number  of  the  orbs  is  increased,  and  their 
thickness  diminished  without  end.  In  like  manner,  by  the  points  of  which 
lines,  surfaces,  and  solids  are  said  to  be  composed,  are  to  be  understood 
equal  particles,  whose  magnitude  is  perfectly  inconsiderable. 

PROPOSITION  LXXIV.     THEOREM  XXXIV. 

The  same  things  supposed,  I  say,  that  a  corpuscle  situate  without  the 

sphere  is  attracted  with  a  force  reciprocally  proportional  to  the  square 

of  its  distance  from  the  centre. 

For  suppose  the  sphere  to  be  divided  into  innumerable  concentric  sphe 
rical  superficies,  and  the  attractions  of  the  corpuscle  arising  from  the  sev 
eral  superficies  will  be  reciprocally  proportional  to  the  square  of  the  dis 
tance  of  the  corpuscle  from  the  centre  of  the  sphere  (by  Prop.  LXXI). 
And,  by  composition,  the  sum  of  those  attractions,  that  is,  the  attraction 
of  the  corpuscle  towards  the  entire  sphere,  will  be  in  the  same  ratio.  Q.E.D. 

COR.  1.  Hence  the  attractions  of  homogeneous  spheres  at  equal  distances 
from  the  centres  will  be  as  the  spheres  themselves.  For  (by  Prop.  LXXII) 
if  the  distances  be  proportional  to  the  diameters  of  the  spheres,  the  forces 
will  be  as  the  diameters.  Let  the  greater  distance  be  diminished  in  that 
ratio ;  and  the  distances  now  being  equal,  the  attraction  will  be  increased 
in  the  duplicate  of  that  ratio ;  and  therefore  will  be  to  the  other  attraction 
in  the  triplicate  of  that  ratio  ;  that  is,  in  the  ratio  of  the  spheres. 

COR.  2.  At  any  distances  whatever  the  attractions  are  as  the  spheres 
applied  to  the  squares  of  the  distances. 

COR.  3.  If  a  corpuscle  placed  without  an  homogeneous  sphere  is  attract 
ed  by  a  force  reciprocally  proportional  to  the  square  of  its  distance  from 
the  centre,  and  the  sphere  consists  of  attractive  particles,  the  force  of  ever y 
particle  will  decrease  in  a  duplicate  ratio  of  the  distance  from  each  particle. 

PROPOSITION  LXXV.     THEOREM  XXXV. 

If  to  the  several  points  of  a  given  sphere  there  tend  equal  centripetal  forces 
decreasing  in  a  duplicate  ratio  of  the  distances  from  the  points  ;  I  say, 
that  another  similar  sphere  will  be  attracted  by  it  with  a  force  recip 
rocally  proportional  to  the  square  of  the  distance  of  the  centres. 
For  the  attraction  of  every  particle  is  reciprocally  as  the  square  of  its 


222  THE    MATHEMATICAL    PRINCIPLES  |  BOOK    L 

distance  from  the  centre  of  the  attracting  sphere  (by  Prop.  LXXIV).  and 
is  therefore  the  same  as  if  that  whole  attracting  force  issued  from  one  sin 
gle  corpuscle  placed  in  the  centre  of  this  sphere.  But  this  attraction  is  as 
great  as  on  the  other  hand  the  attraction  of  the  same  corpuscle  would  be, 
if  that  were  itself  attracted  by  the  several  particles  of  the  attracted  sphere 
with  the  same  force  with  which  they  are  attracted  by  it.  But  that  attrac 
tion  of  the  corpuscle  would  be  (by  Prop.  LXXIV)  reciprocally  propor 
tional  to  the  square  of  its  distance  from  the  centre  of  the  sphere  :  therefore 
the  attraction  of  the  sphere,  equal  thereto,  is  also  in  the  same  ratio.  Q,.E.D. 

COR.  1.  The  attractions  of  spheres  towards  other  homogeneous  spheres 
are  as  the  attracting  spheres  applied  to  the  squares  of  the  distances  of  their 
centres  from  the  centres  of  those  which  they  attract. 

COR.  2.  The  case  is  the  same  when  the  attracted  sphere  does  also  at 
tract.  For  the  several  points  of  the  one  attract  the  several  points  of  the 
other  with  the  same  force  with  which  they  themselves  are  attracted  by  the 
others  again;  and  therefore  since  in  all  attractions  (by  Law  III)  the  at 
tracted  and  attracting  point  are  both  equally  acted  on,  the  force  will  be 
doubled  by  their  mutual  attractions,  the  proportions  remaining. 

COR.  3.  Those  several  truths  demonstrated  above  concerning  the  motion 
of  bodies  about  the  focus  of  the  conic  sections  will  take  place  when  an 
attracting  sphere  is  placed  in  the  focus,  and  the  bodies  move  without  the 
sphere. 

COR.  4.  Those  things  which  were  demonstrated  before  of  the  motion  of 
bodies  about  the  centre  of  the  conic  sections  take  place  when  the  motions 
are  performed  within  the  sphere. 

PROPOSITION  LXXVI.     THEOREM  XXXVI. 

ff  spheres  be  however  dissimilar  (as  to  density  of  matter  and  attractive, 
force]  in  the  same  ratio  onward  from  the  centre  to  the  circumference  ; 
but  every  where  similar,  at  every  given  distance  from  the  centre,  on  all 
sides  round  about  ;  and  the  attractive  force  of  every  point  decreases 
in  the  duplicate  ratio  of  the  distance  of  the  body  attracted  ;  I  say, 
that  the  whole  force  with  which  one  of  these  spheres  attracts  the  oilier 
will  be  reciprocally  proportional  to  the  square  of  the  distance  of  the 
centres. 

Imagine  several  concentric  similar 
spheres,  AB,  CD,  EF,  &c..  the  inner 
most  of  which  added  to  the  outermost 
may  compose  a  matter  more  dense  to 
wards  the  centre,  or  subducted  from 
them  may  leave  the  same  more  lax  and 
rare.  Then,  by  Prop.  LXXV,  these 
sphere?  will  attract  other  similar  con- 


SEC.    XII.]  OF    NATURAL    PHILOSOPHY.  223 

eentric  spheres  GH;  IK,  LM,  &c.,  each  the  other,  with  forces  reciprocally 
proportional  to  the  square  of  the  distance  SP.  And,  by  composition  or 
division,  the  sum  of  all  those  forces,  or  the  excess  of  any  of  them  above 
the  others;  that  is,  the  entire  force  with  which  the  whole  sphere  AB  (com 
posed  of  any  concentric  spheres  or  of  their  differences)  will  attract  the 
whole  sphere  GH  (composed  of  any  concentric  spheres  or  their  differences) 
in  the  same  ratio.  Let  the  number  of  the  concentric  spheres  be  increased 
in  infinitum,  so  that  the  density  of  the  matter  together  with  the  attractive 
force  may,  in  the  progress  from  the  circumference  to  the  centre,  increase  or 
decrease  according  to  any  given  law  ;  and  by  the  addition  of  matter  not  at 
tractive,  let  the  deficient  density  be  supplied,  that  so  the  spheres  may  acquire 
any  form  desired ;  and  the  force  with  which  one  of  these  attracts  the  other 
will  be  still,  by  the  former  reasoning,  in  the  same  ratio  of  the  square  of  the 
distance  inversely.  Q.E.I). 

COR.  I.  Hence  if  many  spheres  of  this  kind,  similar  in  all  respects,  at 
tract  each  other  mutually,  the  accelerative  attractions  of  each  to  each,  at 
any  equal  distances  of  the  centre's,  will  be  as  the  attracting  spheres. 

COR.  2.  And  at  any  unequal  distances,  as  the  attracting  spheres  applied 
to  the  squares  of  the  distances  between  the  centres. 

/'  COR.  3.  The  motive  attractions,  or  the  weights  of  the  spheres  towards 
one  another,  will  be  at  equal  distances  of  the  centres  as  the  attracting  and 
attracted  spheres  conjunctly  ;  that  is,  as  the  products  arising  from  multi 
plying  the  spheres  into  each  other. 

COR.  4.  And  at  unequal  distances,  as  those  products  directly,  and  the 
squares  of  the  distances  between  the  centres  inversely. 

COR.  5.  These  proportions  take  place  also  when  the  attraction  arises 
from  the  attractive  virtue  of  both  spheres  mutually  exerted  upon  each 
other.  For  the  attraction  is  only  doubled  by  the  conjunction  of  the  forces, 
the  proportions  remaining  as  before. 

COR.  6.  If  spheres  of  this  kind  revolve  about  others  at  rest,  each  about 
each  ;  and  the  distances  between  the  centres  of  the  quiescent  and  revolving 
bodies  are  proportional  to  the  diameters  of  the  quiescent  bodies ;  the  peri 
odic  times  will  be  equal. 

COR.  7.  And,  again,  if  the  periodic  times  are  equal,  the  distances  will 
be  proportional  to  the  diameters. 

COR.  8.  All  those  truths  above  demonstrated,  relating  to  the  motions 
'jf  bodies  about  the  foci  of  conic  sections,  will  take  place  when  an  attract 
ing  sphere,  of  any  form  and  condition  like  that  above  described,  is  placed 
in  the  focus. 

COR.  9.  And  also  when  the  revolving  bodies  are  also  attracting  spheres 
Df  any  condition  like  that  above  described. 


224  THE    MATHEMATICAL    PRINCIPLES  [BOOK  I. 

PROPOSITION  LXXVI1.     THEOREM  XXXVII. 

Tf  to  1  he  several  points  of  spheres  there  tend  centripetal  forces  propor 
tional  to  the  distances  of  the  points  from  the  attracted  bodies  ;  I  say, 
that  the  compounded  force  with  which  two  spheres  attract  each  other 
mutually  is  as  the  distance  between  the  centres  of  the  spheres. 

CASE  1.  Let  AEBF  be  a  sphere ;  S  its 
centre .  P  a  corpuscle  attracted  :  PA  SB 
the  axis  of  the  sphere  passing  through  the 
centre  of  the  corpuscle  ;  EF,  ef  two  planes 
cutting  the  sphere,  and  perpendicular  to 
the  axis,  and  equi-distant,  one  on  one  side, 
the  other  on  the  other,  from  the  centre  of 
the  sphere ;  G  and  g-  the  intersections  of 
the  planes  and  the  axis  ;  and  H  any  point  in  the  plane  EF.  The  centri 
petal  force  of  the  point  H  upon  the  corpuscle  P,  exerted  in  the  direction  of 
the  line  PH,  is  as  the  distance  PH ;  and  (by  Cor.  2,  of  the  Laws)  the  same 
exerted  in  the  direction  of  the  line  PG,  or  towards  the .  centre  S,  is  as  the 
length  PG.  Therefore  the  force  of  all  the  points  in  the  plane  EF  (that  is, 
of  that  whole  plane)  by  which  the  corpuscle  P  is  attracted  towards  the 
centre  S  is  as  the  distance  PG  multiplied  by  the  number  of  those  points, 
that  is,  as  the  solid  contained  under  that  plane  EF  and  the  distance  PG. 
And  in  like  manner  the  force  of  the  plane  ef,  by  which  the  corpuscle  P  is 
attracted  towards  the  centre  S,  is  as  that  plane  drawn  into  its  distance  Pg, 
or  as  the  equal  plane  EF  drawn  into  that  distance  Pg* ;  and  the  sum  of  the 
forces  of  both  planes  as  the  plane  EF  drawn  into  the  sum  of  the  distances 
PG  +  P^,  that  is,  as  that  plane  drawn  into  twice  the  distance  PS  of  the 
centre  and  the  corpuscle  ;  that  is,  as  twice  the  plane  EF  drawn  into  the  dis 
tance  PS,  or  as  the  sum  of  the  equal  planes  EF  +  ef  drawn  into  the  same 
distance.  And,  by  a  like  reasoning,  the  forces  of  all  the  planes  in  the 
whole  sphere,  equi-distant  on  each  side  from  the  centre  of  the  sphere,  are 
as  the  sum  of  those  planes  drawn  into  the  distance  PS,  that  is,  as  the 
whole  sphere  and  the  distance  PS  conjunctly.  Q,.E.D. 

CASE  2.  Let  now  the  corpuscle  P  attract  the  sphere  AEBF.  And,  by 
the  same  reasoning,  it  will  appear  that  the  force  with  which  the  sphere  is 
attracted  is  as  the  distance  PS.  Q,.E.D. 

CASE  3.  Imagine  another  sphere  composed  of  innumerable  corpuscles  P  : 
and  because  the  force  with  which  every  corpuscle  is  attracted  is  as  the  dis 
tance  of  the  corpuscle  from  the  centre  of  the  first  sphere,  and  as  the  same 
sphere  conjunctly,  and  is  therefore  the  same  as  if  it  all  proceeded  from  a 
single  corpuscle  situate  in  the  centre  of  the  sphere,  the  entire  force  with 
which  all  the  corpuscles  in  the  second  sphere  are  attracted,  that  is,  with 
which  that  whole  sphere  is  attracted,  will  be  the  same  as  if  that  sphere 


SEC.  Xll.]  OP    NATURAL    PHILOSOPHY.  225 

were  attracted  by  a  force  issuing  from  a  single  corpuscle  in  the  centre  of 
the  first  sphere ;  and  is  therefore  proportional  to  the  distance  between  the 
centres  of  the  spheres.  Q,.E.D. 

CASE  4.  Let  the  spheres  attract  each  other  mutually,  and  the  force  will 
be  doubled,  but  the  proportion  will  remain.  Q..E.D. 

CASE  5.  Let  the  corpuscle  p  be  placed  within  ^- ^\E 

the  sphere  AEBF ;  and  because  the  force  of  the 
plane  ef  upon  the  corpuscle  is  as  the  solid  contain 
ed  under  that  plane  and  the  distance  jog' ;  and  the 
contrary  force  of  the  plane  EF  as  the  solid  con 
tained  under  that  plane  and  the  distance  joG ;  the  ^ 
force  compounded  of  both  will  be  as  the  difference  ** 
of  the  solids,  that  is,  as  the  sum  of  the  equal  planes  drawn  into  half  the 
difference  of  the  distances  ;  that  is,  as  that  sum  drawn  into  joS,  the  distance 
of  the  corpuscle  from  the  centre  of  the  sphere.  And,  by  a  like  reasoning, 
the  attraction  of  all  the  planes  EF,  ef,  throughout  the  whole  sphere,  that 
is,  the  attraction  of  the  whole  sphere,  is  conjunctly  as  the  sum  of  all  the 
planes,  or  as  the  whole  sphere,  and  as  joS,  the  distance  of  the  corpuscle  from 
the  centre  of  the  sphere.  Q.E.D. 

CASE  6.  And  if  there  be  composed  a  new  sphere  out  of  innumerable  cor 
puscles  such  as  jo,  situate  within  the  first  sphere  AEBF,  it  may  be  proved, 
as  before,  that  the  attraction,  whether  single  of  one  sphere  towards  the 
other,  or  mutual  of  both  towards  each  other,  will  be  as  the  distance  joS  of 
the  centres.  Q,  E.D. 

PROPOSITION  LXXVIII.     THEOREM  XXXVIII. 

If  spheres  it*  the  progress  from  the  centre  to  the  circumference  be  hoivMtv 
dissimilar  a->id  unequable,  but  similar  on  every  side  round  about  af  all 
given  distances  from  the  centre  ;  and  the  attractive  force  of  evsrt/ 
point  be  as  the  distance  of  the  attracted  body  ;  I  say,  that  the  entire 
force  with  which  two  spheres  of  this  kind  attract  each  other  mutitallij 
is  proportional  to  the  distance  between  the  centres  of  the  spheres. 
This  is  demonstrated  from  the  foregoing  Proposition,  in  the  same  man 
ner  as  Proposition  LXXVI  was  demonstrated  from  Proposition  LXXY. 

COR.  Those  things  that  were  above  demonstrated  in  Prop.  X  and  LXJV, 
of  the  motion  of  bodies  round  the  centres  of  conic  sections,  take  place  when 
all  the  attractions  are  made  by  the  force  of  sphaerical  bodies  of  the  condi 
tion  above  described,  and  the  attracted  bodies  are  spheres  of  the  same  kind. 

SCHOLIUM. 

i  have  now  explained  the  two  principal  cases  of  attractions;  to  wit, 
when  the  centripetal  forces  decrease  in  a  duplicate  ratio  of  the  distances 
•r  increase  in  a  simple  ratio  of  the  distances,  causing  the  bodies  in  botli 

15 


226  THE    MATHEMATICAL    PRINCIPLES  [BoOK    1 

cases  to  revolve  in  conic  sections,  and  composing  sphaerical  bodies  whose 
centripetal  forces  observe  the  same  law  of  increase  or  decrease  in  the  recess 
from  the  centre  as  the  forces  of  the  particles  themselves  do  ;  which  is  verv 
remarkable.  It  would  be  tedious  to  run  over  the  other  cases,  whose  con 
clusions  are  less  elegant  and  important,  so  particularly  as  I  have  done 
these.  I  choose  rather  to  comprehend  and  determine  them  all  by  one  gen 
eral  method  as  follows. 

LEMMA  XXIX. 

ff  about  the  centre  S  there  be  described  any  circle  as  AEB,  and  about  the 
centre  P  there  be.  also  described  two  circles  EF,  ef,  cutting  the  Jirst  in 
E  and  e,  and  the  line  PS  in  F  and  f ;  and  there  be  let  fall  to  PS  the 
perpendiculars  ED,  ed ;  I  say,  that  if  the  distance  of  the  arcs  EF;  ef 
be  supposed  to  be  infinitely  diminished,  the  last  ratio  of  the  evanscent 
linr  Dd  to  the  evanescent  line  Ff  is  the  same  as  that  of  the  line  PE  to 
the  live  PS. 
For  if  the  line  Pe  cut  the  arc  EF  in  q  ;  and  the  right  line  Ee,  which 


coincides  with  the  evanescent  arc  Ee,  be  produced,  and  meet  the  right  line 
PS  in  T  ;  and  there  be  let  fall  from  S  to  PE  the  perpendicular  SG  ;  then, 
because  of  the  like  triangles  DTE,  </'!>,  DES,  it  will  be  as  Dd  to  Ee  so 
))T  to  TE,  or  DE  to  ES  :  and  because  the  triangles,  Ee?,  ESG  (by  Lem. 
VIII,  and  Cor.  3,  Lem.  VII)  are  similar,  it  will  be  as  Ee  to  eq  or  F/soES 
to  SG  ;  and,  ex  ceqno,  as  Dd  to  Ff  so  DE  to  SG  ;  that  is  (because  of  the 
similar  triangles  PDE;  PGS),  so  is  PE  to  PS.  Q.E.D. 

PROPOSITION  LXXIX.     THEOREM  XXXIX. 

Suppose  a  superficies  as  EFfe  to  have  its  breadth  infinitely  diminished, 
and  to  be  just  vanishing  ;  and  that  the  same  superficies  by  its  revolu- 
tion  round  the  axis  PS  describes  a  spherical  concavo-convex  solid,  to 
the  several  equnJ  particle*  of  which  there  tend  equal  centripetal  forces  ; 
I  soy,  that  the  force  with  which  thit  solid  attracts  a  corpuscle  situate 
in  P  is  in  a  ratio  compounded  of  the  ratio  of  the  solid  DE2  X  Ff  and 
the  ratio  of  the  force  with  which  the  given  particle  in  the  place  Ff 
would  attract  the  same  corpuscle. 
For  if  we  consider,  first,  the  force  of  the  spherical  superficies  FE  which 


SEC.  xn.j 


OF    NATURAL    PHILOSOPHY. 


227 


is  generated  by  the  revolution  of  the  arc  FE, 
and  is  cut  any  where,  as  in  r,  by  the  line</6, 
the  annular  part  of  the  super  J'cies  generated 
by  the  revolution  of  the  arc  rE  will  be  as  the 
lineola  Dd,  the  radius  of  the  sphere  PE  re- 
mainiag  the  same;  as  Archimedes  has  de 
monstrated  in  his  Book  of  the  Sphere  and 
Cylinder.  And  the  force  of  this  super 
ficies  exerted  in  the  direction  of  the  lines  PE 
or  Pr  situate  all  round  in  the  conical  superficies,  will  be  as  this  annular 
superficies  itself;  that  is  as  the  lineola  DC/,  or,  which  is  the  same,  as  the 
rectangle  under  the  given  radius  PE  of  the  sphere  and  the  lineola  DC/  ;  but 
that  force,  exerted  in  the  direction  of  the  line  PS  tending  to  the  centre  S, 
will  be  less  in  the  ratio  PI)  to  PE,  and  therefore  will  be  as  PD  X  DC/. 
Suppose  now  the  line  DF  to  be  divided  into  innumerable  little  equal  par 
ticles,  each  of  which  call  DC/,  and  then  the  superficies  FE  will  be  divided 
into  so  many  equal  annuli,  whose  forces  will  be  as  the  sum  of  all  the  rec 
tangles  PD  X  DC/,  that  is,  as  |PF2  — -  |PD2;  and  therefore  as  DE-. 
Let  now  the  superficies  FE  be  drawn  into  the  altitude  F/;  and  the  force 
of  the  solid  EF/e  exerted  upon  the  corpuscle  P  will  be  as  DE2  X  Ff; 
that  is,  if  the  force  be  given  which  any  given  particle  as  Ff  exerts  upon 
the  corpuscle  P  at  the  distance  PF.  But  if  that  force  be  not  given,  the 
force  of  the  solid  EF/e  will  be  as  the  solid  DE2  X  Ff  and  that  force  not 
given,  conjunctly.  Q.E.D. 

PROPOSITION  LXXX.     THEOREM  XL. 

If  to  the  several  equal  parts  of  a  sphere  ABE  described  about  the  centre 
S  there  tend  equal  centripetal  forces  ;  and  from  the  several  points  I) 
in  the  axis  of  the  sphere  AB  in  which  a  corpuscle,  as  F,  is  placed, 
there  be  erected  the  perpendiculars  DE  meeting  the  sphere  in  E,  and 
if  in  those  perpendiculars  the  lengths  DN  be  taken  as  the  quantity 

DE2  X  PS 

— —-, ,  and  as  th*  force  which  a  particle  of  the  sphere  situate  in, 

the  axis  exerts  at  the  distance  PE  upon  the  corpuscle  P  conjunctly  ;  ] 
say,  that  the  in  hole  force  with  which  the,  corpuscle  P  is  attracted  to 
wards  the  sphere  is  as  the  area  ANB,  comprehended  under  the  axis  of 
the  sphere  AB,  and  the  curve  line  ANB,  the  locus  of  the  point  N. 
For  supposing  the  construction  in  the  last  Lemma  and  Theorem    to 
stand,  conceive  the  axis  of  the  sphere  AB  to  be  divided  into  innumerable 
equal  particles  DC/,  and  the  whole  sphere  to  be  divided  into  so  many  sphe 
rical  concavo-convex  laminae  EF/e  /  and  erect  the  perpendicular  dn.     By 
the  last  Theorem,  the  force  with  which  the  laminas  EF/e  attracts  the  cor 
puscle  P  is  as  DE2  X  Ff  and   the  force  of  one  particle  exerted  at  the 


228 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK    I. 


distance  PE  or  PF,  conjunctly. 
But  (by  the  last  Lemma)  Dd  is  to 
F/  as  PE  to  PS,  and  therefore  F/ 


. 
is  equal  to 


PE 


F/  is  equal  to  Dd  X 


;  and  DE2  X 

DE2  X  PS 
PET~; 


and  therefore  the  force  of  the  la- 

DE2  X  PS 
mina  EF/e  is  as  Do?  X PT?~ 

and  the  force  of  a  particle  exerted  at  the  distance  PF  conjunctly  ;  that  is, 
by  the  supposition,  as  DN  X  D(/7  or  as  the  evanescent  area  DNwrf. 
Therefore  the  forces  of  all  the  lamina)  exerted  upon  the  corpuscle  P  are  as 
all  the  areas  DN//G?,  that  is,  the  whole  force  of  the  sphere  will  be  as  the 
whole  area  ANB.  Q.E.D. 

COR.  1.  Hence  if  the  certripetal  force  tending  to   the  several  particles 

•p)F2  vx  po 
remain  always  the  same  at  all  distances,  and  DN  be  made  as ™ ; 

Jr  Jli 

the  whole  force  with  which  the  corpuscle  is  attracted  by  the  sphere  is  as 
the  area  ANB. 

COR.  2.  If  the  centripetal  force  of  the  particles  be  reciprocally  as  the 

DE2  X   PS 

distance  of  the  corpuscle  attracted  by  it,  and  DN  be  made  as  -— ^^ , 

the  force  with  which  the  corpuscle  P  is  attracted  by  the  whole  sphere  wil] 
be  as  the  area  ANB. 

Cor.  3.  Jf  the  centripetal  force  of  the  particles  be  reciprocally  as  the 
cube  of  the  distance  of  the  corpuscle  attracted  by  it,  and  DN  be  made  as 

T)F2  y  PS 

— --- — — .  the  force  with  which  the  corpuscle  is  attracted  by  the  whole 

sphere  will  be  as  the  area  ANB. 

COR.  4.  And  universally  if  the  centripetal  force  tending  to  the  several 
particles  of  the  sphere  be  supposed  to  be  reciprocally  as  the  quantity  V ; 

DE2  X  PS 

and  D5&  be  made  as     „„   — ^-  ;  the  force  with  which  a  corpuscle  is  at- 

Jr  Jtj   X     ' 

tracted  by  the  whole  sphere  will  be  as  the  area  ANB. 

PROPOSITION  LXXXI.    PROBLEM  XLI. 

T/Le  things  remaining  as  above,  it  is  required  lo  measure  the  area 

ANB. 

From  the  point  P  let  there  be  drawn  the  right  line  PH  touching  the 
sphere  in  H  ;  and  to  the  axis  PAB,  letting  fall  the  perpendicular  HI, 
bisect  PI  in  L;  and  (by  Prop.  XII,  Book  II,  El  em.)  PE2  is  equal  tf 


SEC.  XII.] 


OF    NATURAL    PHILOSOPHY. 


229 


PS3  +  SE2  +  2PSD.    But  because 

the  triangles  SPH,  SHI  are  alike, 

SE2  or  SH2  is  equal  to  the  rectan 

gle  PSI,     Therefore  PE2   is  equal 

to  the  rectangle  contained  under  PS 

and  PS  -f  SI  +  2SD  ;  that  is,  under 

PS  and  2LS  +  2SD  ;  that  is,  under 

PS  and  2LD.     Moreover  DE2   is 

equal  to   SE2  —  SD%  or  SE2  — 

LS2  +  2SLD  —  LD2,  that  is,  2SLD  —  LD2  —  ALB.    For  LS°-  — 

SE2  or  LSa  —  SAa  (by  Prop.  VI,  Book  II,  Elem.)  is  equal  to  the  rectan 

gle  ALB.     Therefore  if  instead  of  DE  2  we  write  2SLD  —  LD  2  —  ALB, 

the  quantity  -™  -  ^-,  which  (by  Cor.  4  of  the  foregoing  Prop.)  is  as 


PE  x 

the  length  of  the  ordinate  DN,  will 
2SLD  x  PS       LD2  X  PS 


now 


resolve  itself  into   three  parts 


ALB  xPS  ... 

-TE3rr~     -pfixT"     -pE^-v-;whereifinsteadofVwewnt 

the  inverse  ratio  of  the  centripetal  force,  and  instead  of  PE  the  mean  pro 
portional  between  PS  and  2LD,  those  three  parts  will  become  ordinates  to 
so  many  curve  lines,  whose  areas  are  discovered  by  the  common  methods. 
Q.E.D. 

EXAMPLE  1.  If  the  centripetal  force  tending  to  the  several  particles  of 
the  sphere  be  reciprocally  as  the  distance ;  instead  of  V  write  PE  the  dis 
tance,  then  2PS  X  LD  for  PE2  ;  and  DN  will  become  as  SL  —  £  LD  — 

•ny  |y     Suppose  DN  equal  to  its  double  2SL  —  LD  — -  — r^  5    an<*   2SL 

the  given  part  of  the  ordinate  drawn  into  the  length  AB  will  describe  the 
rectangular  area  2SL  X  AB ;  and  the  indefinite  part  LD,  drawn  perpen 
dicularly  into  the  same  length  with  a  continued  motion,  in  such  sort  as  in 
its  motion  one  way  or  another  it  may  either  by  increasing  or  decreasing  re- 

LB2-LA2 

main  always  equal  to  the  length  LD,  will  describe  the  area  —  — ^ —  — , 

that  is,  the  area  SL  X  AB ;  which  taken  from  the  former  area  2SL  X 

AB,  leaves  the  area  SL  X  AE.     But  the  third  part  -  ---,  drawn  after  the 

i  lit, 

same  manner  with  a  continued  motion  perpendicularly  into  the  same  length, 

will  describe  the  area  of  an  hyperbola,  which  subducted 

from  the  area  SL  X  AB  will  leave  ANB  the  area  sought. 

Whence  arises    this  construction  of  the  Problem.     At 

the  points,  L,  A,  B,  erect  the  perpendiculars  L/,  Act,  B6; 

making  Aa  equal  to  LB,  and  Bb  equal  to  LA.     Making 

L/  and  LB  asymptotes,  describe  through  the  points  a,  6, 


230 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK   1 


the  hyperbolic  crrve  ab.     And  the  chord  ba  being  drawn,  will  inclose  the 
area  aba  equal  to  the  area  sought  ANB. 

EXAMPLE  2.  If  the  centripetal  force  tending  to  the  several  particles  of 
the  sphere  be  reciprocally  as  the  cube  of  the  distance,  or  (which  is  the  same 

PE3 


thing;  as  that  cube  applied  to  any  given  plane ;  write 
2PS  X  LD  for  PE2 ;    and   DN  will  become  as 


'2AS2 
SL  X  AS2 


for    V,   and 

AS2 


ALB  X  AS2 
2PS  X  LD2 
LSI 


PS  X  LD       2PS 

that  is  (because  PS,  AS,  SI  are  continually  proportional),  as 
ALB  X  SI 


2LD: 
LSI 


If  we  draw  then  these  three  parts  into  th 


length  AB,  the  first  r-pr  will  generate  the  area  of  an  hyperbola  ;  the  sec- 

L-t  \J 

,  ALB  X  SI  .  ALB  X  SI 

ond  iSI  the  area  }  AB  X  SI  ;  the  third      2Ll^  —        area  -  2LA 


,  that  is,  !AB  X  SI.     From  the  first  subduct  the  sum  of  the 
2LB 

second  and  third,  and  there  will  remain  ANB,  the  area  sought.  Whence 
arises  this  construction  of  the  problem.  At  the  points  L,  A,  S,  B,  erect 
the  perpendiculars  L/  Aa  Ss,  Bb,  of  which  suppose  Ss 
equal  to  SI  ;  and  through  the  point  s,  to  the  asymptotes 
L/,  LB,  describe  the  hyperbola  asb  meeting  the 
perpendiculars  Aa,  Bb,  in  a  and  b  ;  and  the  rectangle 
2ASI,  subducted  from  the  hyberbolic  area  AasbB,  will 
B  leave  ANB  the  area  sought. 


..  ,,    .      „ 

EXAMPLE  3.  If  the  centripetal  force  tending  to  the  several  particles  of 
the  spheres  decrease  in  a  quadruplicate  ratio  of  the  distance  from  the  par- 

pT^4  _ 

tides  ;  write  ~|f-  for  V,  then  V  2PS  +  LD  for  PE,  and  DN  will  become 


___ 
V2SI      X 


SI2    X  ALB 

2v2SI 


X 


These  three  parts  drawn  into   the  length  AB,  produce  so  many  areas,  viz. 


J-L 


2SI2  X  SL  .  1 

x^ —  into 


T— r 
LA 


~~~5ot  in*0    V  LB  —  V  LA;  and 
BS12    X    ALB  .          "1  1" 


VLA3     v/LB3 
And  these  after  due  reduction  come 


forth          __ 


SEC.  XII.]  OF    NATURAL    PHILOSOPHY.  2'3\. 

2SI3  4 SI3 

~oj-p     And   these  by  subducting  the  last  from    the   first,  become  -oT~r 

Therefore  the  entire  force  with  ,7hich  the  corpuscle  P  is  attracted  towards 
the  centre  of  the  sphere  is  as-^,   that   is,  reciprocally   as   PS3    X   PJ 

Q.E.I. 

By  the  same  method  one  may  determine  the  attraction  of  a  corpuscle 
situate  within  the  sphere,  but  more  expeditiously  by  the  following  Theorem. 

PROPOSITION  LXXXIL     THEOREM  XLI. 

In  a  sphere  described  about  the  centre  S  with  the  interval  SA,  if  there  be 
taken  SI,  SA,  SP  continually  proportional  ;  !  sat/,  that  the  attraction, 
of  a  corpuscle  within  the  sphere  in  any  place  I  is  to  its  attraction  without 
the  sphere  in  the  place  P  in  a  ratio  compounded  of  the  subduplicate 
ratio  of  IS,  PS,  the  distances  from  the  centre,  and  the  subduplicate 
ratio  of  tJie  centripetal  forces  tending  to  the  centre  in  those  places  P 
and  I. 

As  if  the  centripetal  forces  of  the 
particles  of  the  sphere  be  reciprocally 
;is  the  distances  of  the  corpuscle  at 
tracted  by  them  ;  the  force  with  which 
the  corpuscle  situate  in  I  is  attracted 
by  the  entire  sphere  will  be  to  the 
force  with  which  it  is  attracted  in  P 
in  a  ratio  compounded  of  the  subdu 
plicate  ratio  of  the  distance  SI  to  the  distance  SP,  and  the  subduplicate 
ratio  of  the  centripetal  force  in  the  place  I  arising  from  any  particle  in  the 
centre  to  the  centripetal  force  in  the  place  P  arising  from  the  same  particle  in 
the  centre ;  that  is,  in  the  subduplicate  ratio  of  the  distances  SI,  SP  to  each 
other  reciprocally.  These  two  subduplicate  ratios  compose  the  ratio  of 
equality,  and  therefore  the  attractions  in  I  and  P  produced  by  the  whole 
sphere  are  equal.  By  the  like  calculation,  if  the  forces  of  the  particles  of 
the  sphere  are  reciprocally  in  a  duplicate  ratio  of  the  distances,  it  will  be 
found  that  the  attraction  in  I  is  to  the  attraction  in  P  as  the  distance  SP 
to  the  semi -diameter  SA  of  the  sphere.  If  those  forces  are  reciprocally  in 
a  triplicate  ratio  of  the  distances,  the  attractions  in  I  and  P  will  be  to  each 
other  as  SP2  to  SA3  ;  if  in  a  quadruplicate  ratio,  as  SP3  to  SA3.  There 
fore  since  the  attraction  in  P  was  found  in  this  last  case  to  be  reciprocally 
as  PS 3  X  PI,  the  attraction  in  I  will  be  reciprocally  as  S  A 3  X  PI,  that  is, 
because  S  A 3  is  given  reciprocally  as  PI.  And  the  progression  is  the  same 
in  injinitnm.  The  demonstration  of  this  Theorem  is  as  follows : 

The  things  remaining  as  above  constructed,  and  a  corpuscle  being  in  anj 


332 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK  I. 


place  P.  the  ordinate  DN  was  found  to  be  as 


T)F2  \"  PS 

—  00  ^~\r-      Therefore  if 

I  Cj    X      V 

IE  be  drawn,  that  ordinate  for  any  other  place  of  the  corpuscle,  as  I,  will 

DE2  X  IS 

become  (mutatis  mutandis]  as  ~T~p~rry~-    Suppose  the  centripetalsorces 

flowing  from  any  point  of  the  sphere,  as  E,  to  be  to  each  other  at  the  dis 
tances  IE  and  PE  as  PE'1  to  IE11  (where  the  number  u  denotes  the  index 

DE2  X  PS 

of  the  powers  of  PE  and  IE),  and  those  ordinates  will  become  as  ^p  -  -57^7, 


2     \x    IS 

and  —  ~"  ---  TT7,     whose  ratio  to  each  other  is  as  PS  X  IE  X  IEn  to  IS  X 

IE   X   IE" 

PE  X  PEn.  Because  SI,  SE,  SP  are  in  continued  proportion,  the  tri 
angles  SPE,  SEI  are  alike  ;  and  thence  IE  is  to  PE  as  IS  to  SE  or  SA. 
For  the  ratio  of  IE  to  PE  write  the  ratio  of  IS  to  SA  ;  and  the  ratio  of 
the  ordinates  becomes  that  of  PS  X  IE"  to  SA  X  PEn.  But  the  ratio  of 
PS  to  SA  is  snbduplicate  of  that  of  the  distances  PS,  SI  ;  and  the  ratio  of 
IE"  to  PE1  (because  IE  is  to  PE  as  IS  to  SA)  is  subduplicate  of  that  of 
the  forces  at  the  distances  PS,  IS.  Therefore  the  ordinates,  and  conse 
quently  the  areas  whioifi  the  ordinates  describe,  and  the  attractions  propor 
tional  to  them,  are  in  a  ratio  compounded  of  those  subduplicate  ratios. 
Q.E.D. 

PROPOSITION  LXXXIII.     PROBLEM  XLII. 

To  find  the  force  with  which  a  corpuscle  placed  in  the  centre  of  a  sphere 
is  attracted  towards  any  segment  of  that  sphere  whatsoever. 

Let  P  be  a  body  in  the  centre  of  that  sphere  and 
RBSD  a  segment  thereof  contained  under  the  plane 
RDS,  and  thesphrcrical  superficies  RBS.  Let  DB  be  cut 
in  F  by  a  sphaerical  superficies  EFG  described  from  the 
centre  P,  and  let  the  segment  be  divided  into  the  parts 
_B  BREFGS,  FEDG.  Let  us  suppose  that  segment  to 
be  not  a  purely  mathematical  but  a  physical  superficies, 
having  some,  but  a  perfectly  inconsiderable  thickness. 
*  Let  that  thickness  be  called  O,  and  (by  what  Archi 
medes  has  demonstrated)  that  superficies  will  be  as 
PF  X  DF  X  O.  Let  us  suppose  besides  the  attrac 
tive  forces  of  the  particles  of  the  sphere  to  be  reciprocally  as  that  power  of 
r.he  distances,  of  which  n  is  index  ;  and  the  force  with  which  the  superficies 

DE2  X  O 

EFG  attracts  the  body  P  will  be  (by  Prop.  LXXIX)  as  --       —    that, 


2DF  X  O 
is,  as  ---?—    -,- 


DF2  X  O 
~"~ppn  —  * 


ppn 

the  perpendicular  FN  drawn  into 


SEC.  XJ11.I 


OF    NATURAL    PHILOSOPHY. 


233 


O  be  proportional  to  this  quantity  ;  and  the  curvilinear  area  BDI,  which 
the  ordinate  FN,  drawn  through  the  length  DB  with  a  continued  motion 
will  describe,  will  be  as  the  whole  force  with  which  the  whole  segment 
RBSD  attracts  the  body  P.  Q.E.I. 

PROPOSITION  LXXXIV.     PROBLEM  XLIII. 

To  find  the  force  with  which  a  corpuscle,  placed  without  the  centre  of  a 
sphere  iti  the  axis  of  any  segment,  is  attracted  by  that  segment. 
Let  the  body  P  placed  in.  the  axis  ADB  of 

the  segment  KBK  be  attracted  by  that  seg 
ment.     About  the  centre  P,  with  the  interval 

PE,  let  the  spherical  superficies  EFK  be  de- 

scribed;  and  let  it  divide   the   segment   into 

two  parts  EBKFE  and  EFKDE.     Find  the 

force    of   the    first   of   those  parts    by   Prop. 

LXXXI,  and  the  force  of  the  latter  part  by 

Prop.  LXXXIII,  and  the  sum  of  the  forces  will  be  the  force  of  the  whole 

segment  EBKDE.     Q.E.I. 

SCHOLIUM. 

The  attractions  of  sphaerical  bodies  being  now  explained,  it  comes  next 
in  order  to  treat  of  the  laws  of  attraction  in  other  bodies  consisting  in  like 
manner  of  attractive  particles ;  but  to  treat  of  them  particularly  is  not  neces 
sary  to  my  design.  It  will  be  sufficient  to  subjoin  some  general  proposi 
tions  relating  to  the  forces  of  such  bodies,  and  the  motions  thence  arising, 
because  the  knowledge  of  these  will  be  of  some  little  use  in  philosophical 
inquiries. 


SECTION  XIII. 

Of  the  attractive  forces  of  bodies  which  are  not  of  a  sphcerical  figure. 

PROPOSITION  LXXXV.     THEOREM  XLIL 

If  a  body  be  attracted  by  another,  and  its  attraction  be  vastly  stronger 
when  it  is  contiguous  to  the  attracting  body  than  when  they  are  sepa 
rated  from  one  another  by  a  very  small  interval ;  the  forces  of  the 
particles  of  the  attracting  body  decrease,  in  the  recess  of  the  body  at 
tracted,  in  more  than  a  duplicate  ratio  of  the  distance  of  the  particles. 
For  if  the  forces  decrease  in  a  duplicate  ratio  of  the  distances  from  the 
particles,  the  attraction  towards  a  sphaerical  body  being  (by  Prop.  LXXIV) 
reciprocally  as  the  square  of  the  distance  of  the  attracted  body  from  the 
centre  of  the  sphere,  will  not  be  sensibly  increased  by  the  contact,  and  it 


234  THE    MATHEMATICAL    PRINCIPLES  [BOOK    1 

\vill  be  still  less  increased  by  it,  if  the  attraction,  in  the  recess  of  the  body 
attracted,  decreases  in  a  still  less  proportion.  The  proposition,  therefore, 
is  evident  concerning  attractive  spheres.  And  the  case  is  the  same  of  con 
cave  sphaerical  orbs  attracting  external  bodies.  And  much  more  does  it 
appear  in  orbs  that  attract  bodies  placed  within  them,  because  there  the 
attractions  diffused  through  the  cavities  of  those  orbs  are  (by  Prop.  LXX) 
destroyed  by  contrary  attractions,  and  therefore  have  no  effect  even  in  the 
place  of  contact.  Now  if  from  these  spheres  and  sphoerical  orbs  we  take 
away  any  parts  remote  from  the  place  of  contact,  and  add  new  parts  any 
where  at  pleas  ore,  we  may  change  the  figures  of  the  attractive  bodies  at 
pleasure ;  but  the  parts  added  or  taken  away,  being  remote  from  the  place 
of  contact,  will  cause  no  remarkable  excess  of  the  attraction  arising  from 
the  contact  of  the  two  bodies.  '1  herefore  the  proposition  holds  good  in 
bodies  of  all  figures.  Q.E.I). 

PROPOSITION  LXXXV1.     THEOREM  XLIII. 

If  the  forces  of  the  particles  of  which  an  attractive  body  is  composed  de 
crease,  in.  the  recess  of  the  attractive  body,  in  a  triplicate  or  more  than 
a  triplicate  ratio  of  the  distance  from  the  particles,  the  attraction  will 
be  vastly  stronger  in  the  point  of  contact  than  when  the  attracting  and 
attracted  bodies  are  separated  from  each  other,  though  by  never  so 
small  an  interval. 

For  that  the  attraction  is  infinitely  increased  when  the  attracted  corpus 
cle  comes  to  touch  an  attracting  sphere  of  this  kind,  appears,  by  the  solu 
tion  of  Problem  XLI,  exhibited  in  ££e  second  and  third  Examples.  The 
same  will  also  appear  (by  comparing  those  Examples  and  Theorem  XLI 
together)  of  attractions  of  bodies  made  towards  concavo-convex  orbs,  whether 
the  attracted  bodies  be  placed  without  the  orbs,  or  in  the  cavities  within 
them.  And  by  aiding  to  or  taking  from  those  spheres  and  orbs  any  at 
tractive  matter  any  where  without  the  place  of  contact,  so  that  the  attrac 
tive  bodies  may  receive  any  assigned  figure,  the  Proposition  will  hold  good 
of  all  bodies  universally.  Q.E.D. 

PROPOSITION  LXXXVII.     THEOREM  XI. IV. 

If  two  bodies  similar  to  each  other,  and  consisting  of  matter  equally  at 
tractive  attract  separately  two  corpuscles  proportional  to  those  bodies, 
and  in  a  like  situation  to  them,  the  accelerative  attractions  of  the  cor 
puscles  towards  the  entire  bodies  will  be  as  the  acccleratire  at  tractions 
of  the  corpuscles  towards  particles  of  the  bodies  proportional  to  the 
wholes,  and  alike  situated  in  them. 

For  if  the  bodies  are  divided  into  particles  proportional  to  the  wholes, 
and  alike  situated  in  them,  it  will  be,  as  the  attraction  towards  any  parti 
cle  of  one  of  the  bodies  to  the  attraction  towards  the  correspondent  particle 


SEC.    A  III.]  OF    NATURAL    PHILOSOPHY.  235 

in  the  other  body,  so  are  the  attractions  towards  the  several  particles  of  the 
iirst  body,  to  the  attractions  towards  the  several  correspondent  particles  of 
the  other  body ;  and,  by  composition,  so  is  the  attraction  towards  the  first 
whole  body  to  the  attraction  towards  the  second  whole  body.  Q,.E.U. 

COR.  1 .  Therefore  if,  as  the  distances  of  the  corpuscles  attracted  increase, 
the  attractive  forces  of  the  particles  decrease  in  the  ratio  of  any  power 
of  the  distances,  the  accelerative  attractions  towards  the  whole  bodies  will 
be  as  the  bodies  directly,  and  those  powers  of  the  distances  inversely.  A* 
if  the  forces  of  the  particles  decrease  in  a  duplicate  ratio  of  the  distances 
from  the  corpuscles  attracted,  and  the  bodies  are  as  A 3  and  B 3,  and  there 
fore  both  the  cubic  sides  of  the  bodies,  and  the  distance  of  the  attracted 
corpuscles  from  the  bodies,  are  as  A  and  B  ;  the  accelerative  attractions 

A3         B3 

towards  the  bodies  will  be  as  —  and  — ,  that  is,  as  A  and  B  the  cubic 

sjides  of  those  bodies.  If  the  forces  of  the  particles  decrease  in  a  triplicate 
ratio  of  the  distances  from  the  attracted  corpuscles,  the  accelerative  attrac- 

A3         B3 

tions  towards  the  whole  bodies  will  be  as  —  and  5--,  that  is,  equal.    If  the 

A.  tj 

forces  decrease  in  a  quadruplicate  ratio,  the  attractions  towards  the  bodies 

A3         B3 

will  be  as-—  an^  04'  *^at  is,  reciprocally  as  the  cubic  sides  A  and  B. 

And  so  in  other  cases. 

COR.  2.  Hence,  on  the  other  hand,  from  the  forces  with  which  like  bodies 
attract  corpuscles  similarly  situated,  may  be  collected  the  ratio  of  the  de 
crease  of  the  attractive  forces  of  the  particles  as  the  attracted  corpuscle 
recedes  from  them  ;  if  so  be  that  decrease  is  directly  or  inversely  in  any 
ratio  of  the  distances. 

PROPOSITION  LXXXVIII.     THEOREM  XLV. 

If  the  attractive  forces  of  the  equal  particles  of  any  body  be  as  the  dis 
tance  of  the  places  from  the  particles,  the  force  of  the  whole  body  will 
tend  to  its  centre  of  gravity  ;  and  will  be  the  same  with  the  force  of 
a  globe,  consisting  of  similar  and  equal  matter,  and  having  its  centre 
in  the  centre  of  gravity. 
Let  the  particles  A,  B,  of  the  body  RSTV  at 
tract  any  corpuscle  Z  with  forces  which,  suppos-| 
ing  the  particles  to  be  equal  between  themselves, 
are  as  the  distances  AZ,  BZ ;  but,  if  they  are 
supposed  unequal,  are  as  those   particles   and 
their  distances  AZ,  BZ,  conjunctly,  or  (if  I  may 
go  speak)  as  those  particles  drawn  into  their  dis 
tances  AZ,  BZ  respectively.     And  let  those  forces  be  expressed  by  the 


236  THE    MATHEMATICAL    PRINCIPLES  [BOOK    1. 

contents  u.ider  A  X  AZ,  and  B  X  BZ.  Join  AB,  and  let  it  be  cut  in  G, 
so  that  AG  may  be  to  BG  as  the  particle  B  to  the  particle  A  :  and  G 
will  be  the  common  centre  of  gravity  of  the  particles  A  and  B.  The  force 
A  X  AZ  will  (by  Cor.  2,  of  the  Laws)  be  resolved  into  the  forces  A  X  GZ 
and  A  X  AG ;  and  the  force  B  X  BZ  into  the  forces  B  X  GZ  and  B  X 
BG.  Now  the  forces  A  X  AG  and  B  X  BG,  because  A  is  proportional  to 
B,  and  BG  to  AG,  are  equal,  and  therefore  having  contrary  directions  de 
stroy  one  another.  There  remain  then  the  forces  A  X  GZ  and  B  X  GZ. 
These  tend  from  Z  towards  the  centre  G,  and  compose  the  force  A  +  B 
X  GZ ;  that  is,  the  same  force  as  if  the  attractive  particles  A  and  B  were 
placed  in  their  common  centre  of  gravity  G,  composing  there  a  little  globe. 

By  the  same  reasoning,  if  there  be  added  a  third  particle  G,  and  the 
force  of  it  be  compounded  with  the  force  A  -f  B  X  GZ  tending  to  the  cen 
tre  G,  the  force  thence  arising  will  tend  to  the  common  centre  of  gravity 
of  that  globe  in  G  and  of  the  particle  C  ;  that  is,  to  the  common  centre  oi 
gravity  of  the  three  particles  A,  B,  C  ;  and  will  be  the  same  as  if  that 
globe  and  the  particle  C  were  placed  in  that  common  centre  composing  a 
greater  globe  there ;  and  so  we  may  go  on  in  injinitum.  Therefore 
the  whole  force  of  all  the  particles  of  any  body  whatever  RSTV  is  the 
same  as  if  that  body,  without  removing  its  centre  of  gravity,  were  to  put 
on  the  form  of  a  globe.  Q,.E.D. 

COR.  Hence  the  motion  of  the  attracted  body  Z  will  be  the  same  as  if 
the  attracting  body  RSTV  were  sphaerical ;  and  therefore  if  that  attract 
ing  body  be  either  at  rest,  or  proceed  uniformly  in  a  right  line,  the  body 
attracted  will  move  in  an  ellipsis  having  its  centre  in  the  centre  of  gravity 
of  the  attracting  body. 

PROPOSITION  LXXXIX.     THEOREM  XLVI. 

If  there  be  several  bodies  consisting  of  equal  particles  whose  jorces  are 
as  the  distances  of  the  places  from  each,  the  force  compounded  of  all 
the  forces  by  which  any  corpuscle  is  attracted  will  tend  to  the  common 
centre  of  gravity  of  the  attracting  bodies  ;  and  will  be  the  same  as  if 
those  attracting  bodies,  preserving  their  common  centre  of  gravity, 
should  unite  there,  and  be  formed  into  a  globe. 

This  is  demonstrated  after  the  same  manner  as  the  foregoing  Proposi 
tion. 

COR.  Therefore  the  motion  of  the  attracted  body  will  be  the  same  as  if 
the  attracting  bodies,  preserving  their  common  centre  of  gravity,  should 
unite  there,  and  be  formed  into  a  globe.  And,  therefore,  if  the  common 
centre  of  gravity  of  the  attracting  bodies  be  either  at  rest,  or  proceed  uni 
formly  in  a  right  line,  the  attracted  body  will  move  in  an  ellipsis  having 
Us  centre  in  the  common  centre  of  gravity  of  the  attracting  bodies. 


SEC.     XlII.j  OF    NATURAL    PHILOSOPHY.  237 

PROPOSITION  XC.     PROBLEM  XLIV. 

If  to  the  several  points  of  any  circle  there  tend  equal  centripeta    forces, 
increasing'  or  decreasing  in  any  ratio  of  the  distances  ;  it  is  required 
to  Jin  d  the  force  with  which  a  corpuscle  is  attracted,  that  is,  situate 
any  where  in  a  right  line  which  stands  at  right  angles  to  the  plant 
of  the  circle  at  its  centre. 
Suppose  a  circle  to  be  described  about  the  cen 
tre  A  with  any  interval  AD  in  a  plane  to  which  ; 
the  right  line  AP  is  perpendicular  ;  and  let  it  be 
required  to  find  the  force  with  which  a  corpuscle 
P  is  attracted  towards  the  same.     From  any  point 
E  of  the  circle,  to  the  attracted  corpuscle  P,  let 
there  be  drawn  the  right  line  PE.     In  the  right 
line  PA  take  PF  equal  to  PE,  and  make  a  per- 
pendicular  FK,  erected  at  F,  to  be  as  the  force 
with  which  the  point  E  attracts  the  corpuscle  P. 
And  let  the  curve  line  IKL  be  the  locus  of  the  point  K.     Let  that  cu/,  fe 
meet  the  plane  of  the  circle  in  L.     In  PA  take  PH  equal  to  PD,  and  p/^ct 
the  perpendicular  HI  meeting  that  curve  in  I ;  and  the  attraction  of  the 
corpuscle  P  towards  the  circle  will  be  as  the  area  AHIL  drawn  into  the 
altitude  AP      Q.E.I. 

For  let  there  be  taken  in  AE  a  very  small  line  Ee.  Join  Pe,  and  in  PE, 
PA  take  PC,  Pf  equal  to  Pe.  And  because  the  force,  with  which  any 
point  E  of  the  annulus  described  about  the  centre  A  with  the  interval  AS 
in  the  aforesaid  plane  attracts  to  itself  the  body  P,  is  supposed  to  be  as 
FK ;  and,  therefore,  the  force  with  which  that  point  attracts  the  body  P 

AP  X  FK 

towards  A  is  as  - — ^p ;  and  the  force  with  which  the  whole  annulus 

AP  X  FK 

attracts  tne  body  P  towards  A  is  as  the  annulus  and p^ conjunct- 

ly ;  and  that  annulus  also  is  as  the  rectangle  under  the  radius  AE  aad  the 
breadth  Ee,  and  this  rectangle  (because  PE  and  AE,  Ee  and  CE  are  pro 
portional)  is  equal  to  the  rectangle  PE  X  CE  or  PE  X  F/;  the  force 
*-ith  which  that  annulus  attracts  the  body  P  towards  A  will  be  as  PE  X 

AP  X  FK 
Ff  and pp~~~  conjunctly ;  that  is,  as  the  content  under  F/  X  FK  X 

AP,  or  as  the  area  FKkf  drawn  into  AP.     And  therefore  the  sum  of  the 
forces  with  which  all  the  annuli,  in  the  circle  described  about  the  centre  A 
with  the  interval  AD,  attract  the  body  P  towards  A,  is  as  the  whole  area 
AHIKL  drawn  into  AP.     Q.E.D. 
COR.  1.  Hence  if  the  forces  of  the  points  decrease  in  the  duplicate  ratio 


238 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK    I 


of  the  distances,  that  is,  if  FK  be  as  rfFK,  and  therefore  the  area  AHIKL 


as  p-7  —  p-     ;  the  attraction  of  the  corpuscle  P  towards  the  circle  will 


PA  AH 

be  as  1  —         ;  that  is,  as 


COR.  2.  And  universally  if  the  forces  of  the  points  at  the  distances  D  b( 
reciprocally  as  any  power  Dn  of  the  distances;  that  is,  if  FK  be  as    —  . 


and  therefore  the  area  AHIKL  as 


1 


1 


"  —  l       PH"  — 

1  PA 


, ;    the  attraction 


of  the  corpuscle  P  towards  the  circle  will  be  as 

PA" — 2       PH" — l 

COR.  3.  And  if  the  diameter  of  the  circle  be  increased  in  itifinitum,  and 
the  number  n  be  greater  than  unity  ;  the  attraction  of  the  corpuscle  P  to 
wards  the  whole  infinite  plane  will  be  reciprocally  as  PA"  —  2,  because  the 

PA 

other  term vanishes. 

PROPOSITION  XCI.     PROBLEM  XLV. 

To  find  the  attraction  of  a  corpuscle  situate  in  the  axis  of  a  round  solid, 
to  whose  several  points  there  tend  equal  centripetal  forces  decreasing 
in  any  ratio  of  the  distances  whatsoever. 

Let  the  corpuscle  P,  situate  in  the  axis  AB 
of  the  solid  DECG,  be  attracted  towards  that 
solid.  Let  the  solid  be  cut  by  any  circle  as 
RFS,  perpendicular  to  the  axis ;  and  in  its 
semi-diameter  FS,  in  any  plane  PALKB  pass 
ing  through  the  axis,  let  there  be  taken  (by 
Prop.  XC)  the  length  FK  proportional  to  the 
force  with  which  the  corpuscle  P  is  attracted 
towards  that  circle.  Let  the  locus  of  the  point 
K  be  the  curve  line  LKI,  meeting  the  planes  of  the  outermost  circles  AL 
and  BI  in  L  and  I ;  and  the  attraction  of  the  corpuscle  P  towards  the 
solid  will  be  as  the  area  LABI.  Q..E.I. 

COR.  1.  Hence  if  the  solid  be  a  cylinder  described  by  the  parallelogram 
A  DEB  revolved  about  the  axis  AB,  and  the  centripetal  forces  tending  to 
the  several  points  be  reciprocally  as  the  squares  of  the  distances  from  the 
points  ;  the  attraction  of  the  corpuscle  P  towards  this  cylinder  will  be  as 
AB  —  PE  +  PD.  For  the  ordinate  FK  (by  Cor.  1,  Prop.  XC)  will  be 

PF 

as  1  —     --.     The  part  1  of  this  quantity,  drawn  into  the  length  AB,  de- 


SEC.  XIII. 


OF    NATURAL    PHILOSOPHY 


239 


scribes  the  area  1  X  AB  ;  and  the  other  part 

PF 

,  drawn  into  the  length  PB  describes  the 


ix 


area  1  into  PE  —  AD  (as  may  be  easily 
shewn  from  the  quadrature  of  the  curve 
LKI);  and,  in  like  manner,  the  same  part 
drawn  into  the  length  PA  describes  the  area 

L  into  PD  —  AD.  and  drawn   into  AB,  the 


"At 


G 


Iv 


S 


13 


M 

7J" 


1 


difference  of  PB  and  PA,  describes  1  into  PE  —  PD,  the  difference  of  the 
areas.  From  the  first  content  1  X  AB  take  away  the  last  content  1  into 
PE  —  PD,  and  there  will  remain  the  area  LABI  equal  to  1  into 
AB  —  PE  -h  PD.  Therefore  the  force,  being  proportional  to  this  area, 
is  as  AB  —  PE  +  PD. 

COR.  2.  Hence  also  is  known  the  force 
by  which  a  spheroid  AGBC  attracts  any 
body  P  situate  externally  in  its  axis  AB. 
Let  NKRM  be  a  conic  section  whose  or- 
dinate  KR  perpendicular  to  PE  may  be  \ 
always  equal  to  the  length  of  the  line  PD, 
continually  drawn  to  tlie  point  D  in 
which  that  ordinate  cuts  the  spheroid. 
From  the  vertices  A,  B,  of  the  spheriod, 
let  there  be  erected  to  its  axis  AB  the  perpendiculars  AK,  BM,  respectively 
equal  to  AP.  BP,  and  therefore  meeting  the  conic  section  in  K  and  M;  and 
join  KM  cutting  off  from  it  the  segment  KMRK.  Let  S  be  the  centre  of  the 
spheroid,  and  SC  its  greatest  semi-diameter  :  and  the  force  with  which  the 
spheroid  attracts  the  body  P  will  be  to  the  force  with  which  a  sphere  describ- 

,    ....,,.  ASxCS2-PSxKMRK 

ed  with  the  diameter  AhJ  attracts  the  same  body  as prrr ^ r-= 

1  o    -f-  Go2  — Ao 

AS3 
is  to  fkT^,.     And  by  a  calculation  founded  on  the  same  principles  may  be 


found  the  forces  of  the  segments  of  the  spheroid. 

COR.  3.  If  the  corpuscle  be  placed  within  the  spheroid  and  in  its  axis, 
the  attraction  will  be  as  its  distance  from  the  centre.  This  may  be  easily 
collected  from  the  following  reasoning,  whether 
the  particle  be  in  the  axis  or  in  any  other  given 
diameter.  Let  AGOF  be  an  attracting  sphe 
roid,  S  its  centre,  and  P  the  body  attracted. 
Through  the  body  P  let  there  be  drawn  the 
semi-diameter  SPA,  and  two  right  lines  DE, 
FC  meeting  the  spheroid  in  1)  and  E,  F  and 
G  ;  and  let  PCM,  HLN  be  the  superficies  of 


240  THE    MATHEMATICAL    PRINCIPLE*  ffioOK    1. 

two  interior  spheroids  similar  and  concentrical  to  the  exterior,  the  first  of 
which  passes  through  the  body  P.  and  cuts  the  right  lines  DE,  FG  in  B 
and  C  ;  arid  the  latter  cuts  the  same  right  lines  in  H  and  I,  K  and  L. 
I  ,et  the  spheroids  have  all  one  common  axis,  and  the  parts  of  the  right 
lines  intercepted  on  both  sides  DP  and  BE,  FP  and  CG,  DH  and  IE,  FK 
and  LG,  will  be  mutually  equal;  because  the  right  lines  DE.  PB,  and  HI. 
are  bisected  in  the  same  point,  as  are  also  the  right  lines  FG,  PC,  and  KL. 
Conceive  now  DPF.  EPG  to  represent  opposite  cones  described  with  the 
infmitely  small  vertical  angles  DPF,  EPG,  and  the  lines  DH,  El  to  be 
infinitely  small  also.  Then  the  particles  of  the  cones  DHKF,  GLIE,  cut 
off  by  the  spheroidical  superficies,  by  reason  of  the  equality  of  the  lines  DH 
and  ET;  will  be  to  one  another  as  the  squares  of  the  distances  from  the  body 
P,  and  will  therefore  attract  that  corpuscle  equally.  And  by  a  like  rea 
soning  if  the  spaces  DPF,  EGCB  be  divided  into  particles  by  the  superfi 
cies  of  innumerable  similar  spheroids  concentric  to  the  former  and  having 

J.  O 

one  common  axis,  all  these  particles  will  equally  attract  on  both  sides  the 
body  P  towards  contrary  parts.  Therefore  the  forces  of  the  cone  DPF. 
and  of  the  conic  segment  EGCB,  are  equal,  and  by  their  contrariety  de 
stroy  each  other.  And  the  case  is  the  same  of  the  forces  of  all  the  matter 
that  lies  without  the  interior  spheroid  PCBM.  Therefore  the  body  P  is 
attracted  by  the  interior  spheroid  PCBM  alone,  and  therefore  (by  Cor.  3, 
Prop.  1 , XXII)  its  attraction  is  to  the  force  with  which  the  body  A  is  at 
tracted  by  the  whole  spheroid  AGOD  as  the  distance  PS  to  the  distance 
AS.  Q.E.D. 

PROPOSITION  XCII.     PROBLEM  XLVI. 

An  attracting  body  being  given,  it  is  required  to  find  the  ratio  of  the  de 
crease  of  the  centripetal  forces  tending  to  its  several  points. 
The  body  given  must  be  formed  into  a  sphere,  a  cylinder,  or  some  regu 
lar  figure,  whose  law  of  attraction  answering  to  any  ratio  of  decrease  may 
be  found  by  Prop.  LXXX,  LXXXI,  and  XCI.     Then,  by  experiments, 
the  force  of  the  attractions  must  be  found  at  several  distances,  and  the  law 
of  attraction  towards  the  whole,  made  known  by  that  means,  will  give 
the  ratio  of  the  decrease  of  the  forces  of  the  several  parts  ;  which  was  to 
be  found. 

PROPOSITION  XCIII.     THEOREM  XLVII. 

If  a  solid  be  plane  on  one  side,  and  infinitely  extended  on  all  otljer  sides, 
and  consist  of  equal  particles  equally  attractive,  iv  hose  forces  decrease, 
in  the  recess  from  the  solid,  in  the  ratio  of  any  power  greater  than  the 
square  of  the  distances  ;  and  a  corpuscle  placed  towards  eit far  part  of 
the  plane  is  attracted  by  the  force  of  the  whole  solid  ;  I  say  that  tfie 
attractive  force  of  the  whole  solid,  in  the  recess  from  its  platw  superfi- 


XIILj 


OF    NATURAL    PHILOSOPHY". 


241 


n 


H 


m 


G 


ties,  will  decrease  in  the  ratio  of  a  power  whose  side  is  the  distance  oj 

the  corpuscle  from  the  plane,  and  its  index  less  by  3  than  the  index  oj 

the  power  of  the  distances. 

CASE  1.  Let  LG/be  the  plane  by  which 
the  solid  is  terminated.  Let  the  solid 
lie  on  that  hand  of  the  plane  that  is  to 
wards  I,  and  let  it  be  resolved  into  in- _. 
numerable  planes  mHM,  //IN,  oKO, 
(fee.,  parallel  to  GL.  And  first  let  the 
attracted  body  C  be  placed  without  the 
solid.  Let  there  be  drawn  CGHI  per 
pendicular  to  those  innumerable  planes, 
and  let  the  attractive  forces  of  the  points  of  the  solid  decrease  in  the  ratio 
of  a  power  of  the  distances  whose  index  is  the  number  n  not  less  than  3. 
Therefore  (by  Cor.  3,  Prop.  XC)  the  force  with  which  any  plane  mHM 
attracts  the  point  C  is  reciprocally  as  CHn— 2.  In  the  plane  mHM  take  the 
length  HM  reciprocally  proportional  to  CH1— 2,  and  that  force  will  be  as 
HM.  In  like  manner  in  the  several  planes  /GL,  //,TN,  oKO,  (fee.,  take  the 
lengths  GL,  IN,  KO,  (fee.,  reciprocally  proportional  to  CGn—  2,  CI1—  2, 
CKn  — 2,  (fee.,  and  the  forces  of  those  planes  will  bs  as  the  lengths  so  taken, 
and  therefore  the  sum  of  the  forces  as  the  sum  of  the  lengths,  that  is,  the 
force  of  the  Avhole  solid  as  the  area  GLOK  produced  infinitely  towards 
OK.  But  that  area  (by  the  known  methods  of  quadratures)  is  reciprocally 
as  CGn— 3,  and  therefore  the  force  of  the  whole  solid  is  reciprocally  as 
CG"-3.  Q.E.D. 

CASE  2.  Let  ttecorpuscleC  be  now  placed  on  that 
hand  of  the  plane  /GL  that  is  within  the  solid, 
and  take  the  distance  CK  equal  to  the  distance 
CG.  And  the  part  of  the  solid  LG/oKO  termi 
nated  by  the  parallel  planes  /GL,  oKO,  will  at 
tract  the  corpuscle  C,  situate  in  the  middle,  neither 
one  way  nor  another,  the  contrary  actions  of  the 
opposite  points  destroying  one  another  by  reason  of 
their  equality.  Therefore  the  corpuscle  C  is  attracted  by  the  force  only 
of  the  solid  situate  beyond  the  plane  OK.  But  this  force  (by  Case  1)  is 
reciprocally  as  CKn—3,  that  is,  (because  CG,  CK  are  equal)  reciprocally  as 
CG"— 3.  Q,.E.D. 

COR.  1.  Hence  if  the  solid  LGIN  be  terminated  on  each  sitfe  by  two  in 
finite  parallel  places  LG,  IN,  its  attractive  force  is  known,  subducting 
from  the  attractive  force  of  the  whole  infinite  solid  LGKO  the  attractive 
force  of  the  more  distant  part  NIKO  infinitely  produced  towards  KO. 

COR.  2.  If  the  more  distant  part  of  this  solid  be  rejected,  because  its  at 
traction  compared  with  the  attraction  of  the  nearer  part  is  inconsiderable, 

16 


0  N 

K     1 
0 

C 

242  THE    MATHEMATICAL    PRINCIPLES  [BOOK    1 

the  attraction  of  that  nearer  part  will,  as  the  distance  increases,  decrease 
nearly  in  the  ratio  of  the  power  CGn—  3. 

Con.  3.  And  hence  if  any  finite  body,  plane  on  one  side,  attract  a  cor 
puscle  situate  over  against  the  middle  of  that  plane,  and  the  distance  between 
the  corpuscle  and  the  plane  compared  with  the  dimensions  of  the  attracting 
body  be  extremely  small  ;  and  the  attracting  body  consist  of  homogeneous 
particles,  whose  attractive  forces  decrease  in  the  ratio  of  any  power  of  the 
distances  greater  than  the  quadruplicate ;  the  attractive  force  of  the  whole 
body  will  decrease  very  nearly  in  the  ratio  of  a  power  whose  side  is  that 
very  small  distance,  and  the  index  less  by  3  than  the  index  of  the  former 
power.  This  assertion  does  not  hold  good,  however,  of  a  body  consisting 
of  particles  whose  attractive  forces  decrease  in  the  ratio  of  the  triplicate 
power  of  the  distances  ;  because,  in  that  case,  the  attraction  of  the  remoter 
part  of  the  infinite  body  in  the  second  Corollary  is  always  infinitely  greater 
than  the  attraction  of  the  nearer  part. 


SCHOLIUM. 

If  a  body  is  attracted  perpendicularly  towards  a  given  plane,  and  from 
the  law  of  attraction  given,  the  motion  of  the  body  be  required  ;  the  Pro 
blem  will  be  solved  by  seeking  (by  Prop.  XXXIX)  the  motion  of  the  body 
descending  in  a  right  line  towards  that  plane,  and  (by  Cor.  2,  of  the  Laws) 
compounding  that  motion  with  an  uniform  motion  performed  in  the  direc 
tion  of  lines  parallel  to  that  plane.  And,  on  the  contrary,  if  there  be  re 
quired  the  law  of  the  attraction  tending  towards  the  plane  in  perpendicu 
lar  directions,  by  which  the  body  may  be  caused  to  move  in  any  given 
curve  line,  the  Problem  will  be  solved  by  working  after  the  manner  of  the 
third  Problem. 

But  the  operations  may  be  contracted  by  resolving  the  ordinates  into 
converging  series.  As  if  to  a  base  A  the  length  B  be  ordinately  ap 
plied  in  any  given  angle,  and  that  length  be  as  any  power  of  the  base 

A^  ;  and  there  be  sought  the  force  with  which  a  body,  either  attracted  to 
wards  the  base  or  driven  from  it  in  the  direction  of  that  ordinate,  may  be 
caused  to  move  in  the  curre  line  which  that  ordinate  always  describes  with 
its  superior  extremity ;  I  suppose  the  base  to  be  increased  by  a  very  small 

,m  m 

part  O,  and  I  resolve  the  ordinate  A  -f  Ol^  into  an  infinite  series  A-  -f 

!!L  OA  ^  +  '^-^---  OOA  — ;-  &c.,  and  I  suppose  the  force  proper- 
11  £1111 

tional  to  the  term  of  this  series  in  which  O  is  of  two  dimensions,  that  is, 
to  the  term  -  -  OOA  ^YT,  Therefore  the  force  sought  is  aa 


SEC.  XIV.J  OF    NATURAL    PHILOSOPHY.  2M 

mm  —  mn      m  —  2n  ....  mm  —  mn      m  —  2n 

A  — 7, — ,  or,  which   is   the  same  thinor,  as L>     m    . 

nn  nn 

As  if  the  ordinate  describe  a  parabola,  m  being  —  2,  and  n  =  1,  the  force 
will  be  as  the  given  quantity  2B°,  and  therefore  is  given.  Therefore  with 
a  given  force  the  body  will  move  in  a  parabola,  as  Galileo  has  demon 
strated.  If  the  ordinate  describe  an  hyperbola,  m  being  =  0  —  1,  and  n 
—  1,  the  force  will  be  as  2  A  3  or  2B 3 ;  and  therefore  a  force  which  is  as  the 
cube  of  the  ordinate  will  cause  the  body  to  move  in  an  hyperbola.  But 
leaving  this  kind  of  propositions,  I  shall  go  on  to  some  others  relating  to 
motion  which  I  have  fiot  yet  touched  upon. 


SECTION  XIV. 

Of  the  motion  of  very  small  bodies  when  agitated  by  centripetal  forces 
tending  to  the  several  parts  of  any  very  great  body. 

PROPOSITION  XCIV.     THEOREM  XLVIII. 

If  two  similar  mediums  be  separated  from  each  other  by  a  space  termi 
nated  on  both  sides  by  parallel  planes,  and  a  body  in  its  passage 
through  that  space  be  attracted  or  impelled  perpendicularly  towards 
either  of  those  mediums,  and  not  agitated  or  hindered  by  any  other 
force  ;  and  the  attraction  be  every  where  the  same  at  equal  distances 
from  either  plane,  taken  towards  the  same  hand  of  the  plane  ;  I  say, 
that  the  sine  of  incidence  upon  either  plane  will  be  to  the  sine  of  emcr 
gence  from  the  other  plane  in  a  given  ratio. 
CASE  1.  Let  Aa  and  B6  be  two  parallel  planes, 
and  let  the  body  light  upon  the  first  plane  Aa  in 
the  direction  of  the  line  GH,  and  in  its  whole 
passage  through  the  intermediate  space  let  it  be 
attracted  or  impelled  towards  the  medium  of  in 
cidence,  and  by  that  action  let  it  be  made  to  de 
scribe  a  curve  line  HI,  and  let  it  emerge  in  the  di 
rection  of  the  line  IK.     Let  there  be  erected  IM 
perpendicular  to  Eb  the  plane  of  emergence,  and 
meeting  the  line  of  incidence  GH  prolonged  in  M,  and  the  plane  of  inci 
dence  Aa  in  R ;  and  let  the  line  of  emergence  KI  be  produced  and  meet 
HM  in  L.     About  the  centre  L,  with  the  interval  LI,  let  a  circle  be  de 
scribed  cutting  both  HM  in  P  and  Q,  and  MI  produced  in  N ;  and,  first, 
if  the  attraction  or  impulse  be  supposed  uniform,  the  curve  HI  (by  what 
Galileo  has  demonstrated)  be  a  parabola,  whose  property  is  that  of  a  roc- 


THE    MATHEMATICAL    PRINCIPLES  [BoOK    1 

tangle  under  its  given  latiis  rectum  and  the  line  IM  is  equal  to  the  squarrf 
cf  HM  ;  and  moreover  the  line  HM  will  be  bisected  in  L.  Whence  if  to 
MI  there  be  let  fall  the  perpendicular  LO,  MO,  OR  will  be  equal;  and 
adding  the  equal  lines  ON,  OI,  the  wholes  MN,  IR  will  be  equal  also. 
Therefore  since  IR  is  given,  MN  is  also  given,  and  the  rectangle  NMI  is 
to  the  rectangle  under  the  latus  rectum  and  IM,  that  is,  to  HMa  in  a  given 
ratio.  But  the  rectangle  NMI  is  equal  to  the  rectangle  PMQ,,  that  is,  to 
the  difference  of  the  squares  ML2,  and  PL2  or  LI2  ;  and  HM2  hath  a  given 
ratio  to  its  fourth  part  ML2;  therefore  the  ratio  of  ML2 —  LI2  to  ML2  is  given, 
and  by  conversion  the  ratio  of  LI2  to  ML ',  and  its  subduplicate,  theratrio 
of  LI  to  ML.  But  in  every  triangle,  as  LMI,  the  sines  jf  the  angles  are 
proportional  to  the  opposite  sides.  Therefore  the  ratio  of  the  sine  of  the 
angle  of  incidence  LMR  to  the  sine  of  the  angle  of  emergence  LIR  is 
given.  QJE.lr). 

CASE  2.  Let  now  the  body  pass  successively  through  several  spaces  ter 
minated  with  parallel  planes  Aa/>B,  B6cC,  &c.,  and  let  it  be  acted  on  by  a 
\       .  force  which  is  uniform  in  each  of  them  separ- 

\ a  ately,  but  different  in  the  different  spaces ;  and 

B  \  fr  by  what  was  just  demonstrated,  the  sine  of  the 

c  ^^  c  angle  of  incidence  on  the  first  plane  Aa  is  to 

the  sine  of  emergence  from  the  second  plane  Bb 

in  a  given  ratio ;  and  this  sine  of  incidence  upon  the  second  plane  Bb  will 
be  to  the  sine  of  emergence  from  the  third  plane  Cc  in  a  given  ratio ;  and 
this  sine  to  the  sine  of  emergence  from  the  fourth  plane  Dd  in  a  given  ra 
tio  ;  and  so  on  in  infinitum  ;  and,  by  equality,  the  sine  of  incidence  on 
the  first  plane  to  the  sine  of  emergence  from  the  last  plane  in  a  given  ratio. 
I  ,et  now  the  intervals  of  the  planes  be  diminished,  and  their  number  be  in 
finitely  increased,  so  that  the  action  of  attraction  or  impulse,  exerted  accord 
ing  to  any  assigned  law,  may  become  continual,  and  the  ratio  of  the  sine  of 
incidence  on  the  first  plane  to  the  sine  of  emergence  from  the  last  plane 
being  all  along  given,  will  be  given  then  also.  QJE.D. 

PROPOSITION  XCV.     THEOREM  XLIX. 

The  same  thing's  being  supposed,  I  say,  that  the  velocity  of  the  body  be 
fore  its  incidence  is  to  its  velocity  after  emergence  as  the  sine  of  emer 
gence  to  the  sine  of  incid  nee. 

Make  AH  and  Id  equal,  and  erect  the  perpen- 
diculars  AG,  dK  meeting  the  lines  of  incidence 
and  emergence  GH,  IK,  in  G  and  K.     In  GH 
--«  take  TH  equal  to  IK,  and  to  the  plane  Aa  let 
^  fall  a  perpendicular  TV.     And  (by  Cor.  2  of  the 
|x^  I        Laws  of  Motion)  let  the  motion  of  the  body  be 
jv-      resolved  into  two,  one  perpendicular  to  the  planes 


SEC.    X1V.J  OF    NATURAL    PHILOSOPHY.  245 

Aa,  Bb,  Cc,  &c,  and  another  parallel  to  them.  The  force  of  attraction  or 
impulse,  acting  in  directions  perpendicular  to  those  planes,  does  not  at  all 
alter  the  motion  in  parallel  directions ;  and  therefore  the  body  proceeding 
with  this  motion  will  in  equal  times  go  through  those  equal  parallel  inter 
vals  that  lie  between  the  line  AG  and  the  point  H,  and  between  the  point 
I  and  the  line  dK ;  that  is,  they  will  describe  the  lines  GH,  IK  in  equal 
times.  Therefore  the  velocity  before  incidence  is  to  the  velocity  after 
emergence  as  GH  to  IK  or  TH,  that  is,  as  AH  or  Id  to  vH,  that  is  (sup 
posing  TH  or  IK  radius),  as  the  sine  of  emergence  to  the  sine  of  inci 
dence.  Q.E.D. 

PROPOSITION  XCVL     THEOREM  L. 

The  same  things  being  supposed,  and  that  the  motion  before  incidence  is 
swifter  than  afterwards  ;  1  sat/,  lhat  if  the  line  of  incidence  be  in 
clined  continually,  the  body  will  be  at  last  reflected,  and  the  angle  of 
reflexion  will  be  equal  to  the  angle  of  incidence. 

For  conceive  the  body  passing  between  the  parallel  planes  Aa,  Bb,  Cc, 
&c.,  to  describe  parabolic  arcs  as  above; 
and  let  those  arcs  be  HP,  PQ,  QR,  &c. 
And  let  the  obliquity  of  the  line  of  inci-  g 
dence  GH  to  the  first  plane  Aa  be  such  rc~  £ 

that  the  sine  of  incidence  may  be  to  the  radius  of  the  circle  whose  sine  it  is, 
in  the  same  ratio  which  the  same  sine  of  incidence  hath  to  the  sine  of  emer 
gence  from  the  plane  Dd  into  the  space  DefeE  ;  and  because  the  sine  of 
emergence  is  now  become  equal  to  radius,  the  angle  of  emergence  will  be  a 
right  one,  and  therefore  the  line  of  emergence  will  coincide  with  the  plane 
Dd.  Let  the  body  come  to  this  plane  in  the  point  R  ;  and  because  the 
line  of  emergence  coincides  with  that  plane,  it  is  manifest  that  the  body  can 
proceed  no  farther  towards  the  plane  Ee.  But  neither  can  it  proceed  in  the 
line  of  emergence  Rd;  because  it  is  perpetually  attracted  or  impelled  towards 
the  medium  of  incidence.  It  will  return,  therefore,  between  the  planes  Cc, 
Dd,  describing  an  arc  of  a  parabola  Q,R</,  whose  principal  vertex  (by  what 
Galileo  has  demonstrated)  is  in  R,  cutting  the  plane  Or  in  the  same  angle 
at  q,  that  it  did  before  at  Q, ;  then  going  on  in  the  parabolic  arcs  qp,  ph, 
&c.,  similar  and  equal  to  the  former  arcs  QP,  PH,  &c.,  it  will  cut  the  rest 
of  the  planes  in  the  same  angles  at  p,  h,  (fee.,  as  it  did  before  in  P,  H,  (fee., 
and  will  emerge  at  last  with  the  same  obliquity  at  h  with  which  it  first 
impinged  on  that  plane  at  H.  Conceive  now  the  intervals  of  the  planes 
Aa,  Bb,  Cc,  Dd,  Ee,  (fee.,  to  be  infinitely  diminished,  and  the  number  in 
finitely  increased,  so  that  the  action  of  attraction  or  impulse,  exerted  ac 
cording  to  any  assigned  law,  may  become  continual;  and,  the  angle  of 
emergence  remaining  all  alor  g  equal  to  the  angle  of  incidence,  will  be 
equal  to  the  same  also  at  last.  Q.E.D. 


246  THE    MATHEMATICAL    PRINCIPLES  IBoOK  1 

SCHOLIUM. 

These  attractions  bear  a  great  resemblance  to  the  reflexions  and  refrac 
tions  of  light  made  in  a  given  ratio  of  the  secants,  as  was  discovered  h} 
Siiellius  ;  and  consequently  in  a  given  ratio  of  the  sines,  as  was  exhibited 
by  Hes  Cortes.  For  it  is  now  certain  from  the  phenomena  of  Jupiter's 
^satellites,  confirmed  by  the  observations  of  different  astronomers,  that  light 
is  propagated  in  succession,  and  requires  about  seven  or  eight  minutes  to 
travel  from  the  sun  to  the  earth.  Moreover,  the  rays  of  light  that  are  in 
our  air  (as  lately  was  discovered  by  Grimaldus,  by  the  admission  of  light 
into  a  dark  room  through  a  small  hole,  which  1  have  also  tried)  in  their 
passage  near  the  angles  of  bodies,  whether  transparent  or  opaque  (such  aa 
the  circular  and  rectangular  edges  of  gold,  silver  and  brass  coins,  or  of 
knives,  or  broken  pieces  of  stone  or  glass),  are  bent  or  inflected  round  those 
bodies  as  if  they  were  attracted  to  them  ;  and  those  rays  which  in  their 
passage  come  nearest  to  the  bodies  are  the  most  inflected,  as  if  they  were 
most  attracted  :  which  tiling  I  myself  have  also  carefully  observed.  And 
those  which  pass  at  greater  distances  are  less  inflected ;  and  those  at  still 
greater  distances  are  a  little  inflected  the  contrary  way,  and  form  three 
fringes  of  colours.  In  the  figure  5  represents  the  edge  of  a  knife,  or  any 


tl      \ 

/ 

K",     \ 

/ 

^•::^ 

x  -^»  .  v 

^--:<*, 

-f:::r;^c  :.-/ 

N  «>V  J 


V          U 

W~"~a~"  "~a  C:    O    la 


kind  of  wedge  AsB  :  and  gowog,fmnif,emtme,  dlsld,  are  rays  inflected  to 
wards  the  knife  in  the  arcs  owo,  nvn,  mtm,  Isl  ;  which  inflection  is  greater 
or  less  according;  to  their  distance  from  the  knife.  Now  since  this  inflec 
tion  of  the  rays  is  performed  in  the  air  without  the  knife,  it  follows  that  the 
rays  which  fall  upon  the  knife  are  first  inflected  in  the  air  before  they  touch 
the  knife.  And  the  case  is  the  same  of  the  rays  falling  upon  glass.  The 
refraction,  therefore,  is  made  not  in  the  point  of  incidence,  but  gradually,  by 
a  continual  inflection  of  the  rays ;  which  is  done  partly  in  the  air  before  they 
touch  the  glass,  partly  (if  [  mistake  not)  within  the  glass,  after  they  have 
entered  it ;  as  is  represented  in  the  rays  ckzc,  bujb^  ahxa,  falling  upon  r, 
q,  p,  and  inflected  between  k  and  z,  i  and  y,  h  and  x.  Therefore  because 
of  the  analogy  there  is  between  the  propagation  of  the  rays  f  light  and  the 
motion  of  bodies,  I  thought  it  not  amiss  to  add  the  followi  »g  Propositions 
far  optical  uses  ;  not  at  all.  considering  the  nature  of  the  rays  of  .light,  or 
inquiring  whether  they  are  bodies  or  not ;  but  only  determining  the  tra 
jectories  of  bodies  which  are  extremely  like  the  trajectories  of  the  rays. 


SEC.    XIV.]  OF    NATURAL    PHILOSOPHY.  247 

PROPOSITION  XCVII.     PROBT.-EM  XL  VII. 
Supposing  t/w  sine  of  incidence  upon  any  superficies  to  be  in  a  given  ra 

tio  to  the  sine  of  emergence  ;  and  that  tha  inflection  of  t/ts  paths  of 

those  bodies  near  that  superficies  is  performed  in  a  very  short  space, 

which  may  be  considered  as  a  point  ;  it  is  required  to  determine  suck 

a  superficies  as  may  cause  all  the  corpuscles  issuing  from  any  one 

given  place  to  converge  to  another  given  place. 

Let  A  be  the  place  from  whence  the  cor-  E 

puscles  diverge  ;  B  the  place  to  which  they 

should  converge  ;  CDE  the  curve  line  which 

by  its  revolution  round  the  axis  AB  describes  .  /C 

the  superficies  sought  ;  D,  E,  any  two  points  of  that  curve  ;  and  EF,  EG, 

perpendiculars  let  fall  on  the  paths  of  the  bodies  AD,  DB.     Let  the  point 

D  approach  to  and  coalesce  with  the  point  E  ;  and  the  ultimate  ratio  of 

the  line  DF  by  which  AD  is  increased,  to  the  line  DG  by  which   DB  is 

diminished,  will  be  the  same  as  that  of  the  sine  of  incidence  to  the  sine  of 

emergence      Therefore  the  ratio  of  the  increment  of  the  line  AD  to  the 

decrement  of  the  line  DB  is  given:  and  therefore  if  in  the  axis  AB  there 

be  taken  any  where  the  point  C   through  which  the  curve  CDE  must 

pass,  and  CM  the  increment  of  AC  be  taken  in  that  given  ratio  to  CN 

the  decrement  of  BC,  and  from  the  centres  A,  B,  with  the  intervals  AM, 

BN,  there  be  described  two  circles  cutting  each  other  in  D  ;  that  point  D 

will  touch  the  curve  sought  CDE,  and,  by  touching  it  any  where  at  pleasure, 

will  determine  that  curve.    Q.E.I. 

COR.  1.  By  causing  the  point  A  or  B  to  go  off  sometimes  in  infinitum, 
and  sometimes  to  move  towards  other  parts  of  the  point  C,  will  bo  obtain 
ed  all  those  figures  which  Cartesins  has  exhibited  in  his  Optics  and  Geom 
etry  relating  to  refractions.  The  invention  of  which  Cartcsius  having 
thought  fit  to  conceal,  is  here  laid  open  in  this  Proposition. 

COR.  2.  If  a  body  lighting  on  any  superfi 
cies  CD  in  the  direction  of  a  ri^ht  line  AD,  Qj- 

O  \    » 

drawn  according  to  any  law,  should  emerge 
in  the  direction  of  another  right  line  DK  ; 
and  from  the  point  C  there  be  drawn  curve 


lines  CP,  CQ,  always  perpendicular  to  AD,  DK  ;  the  increments  of  the 
lines  PD,  QD,  and  therefore  the  lines  themselves  PD,  Q.D,  generated  by 
those  increments,  will  be  as  the  sines  of  incidence  and  emergence  to  each 
other,  and  e  contra. 

PROPOSITION  XCVIII.     PROBLEM  XLVIII. 

The  same  things  supposed  ;  if  round  the  axis  AB  any  attractive  super 
ficies  be  described  as  CD,  regular  or  irregular,  through  which  the  bo 
dies  issuing  from  the  given  place  A  must  pass  ;  it  is  required  to  find 


THE    MATHEMATICAL    PRINCIPLES.  [BOOK    J 

a  second  attractive  superficies  EF,  which  may  make  those  bodies  con 
verge  to  a  given  place  B. 

Let  a  line  joining  AB  cut 
the  first  superficies  in  C  and 
the  second  in  E,  the  point  D 
being  taken  any  how  at  plea 
sure.  And  supposing  the 
f  sine  of  incidence  on  the  first 
superficies  to  the  sine  of 
emergence  from  the  same,  and  the  sine  of  emergence  from  the  second  super 
ficies  to  the  sine  of  incidence  on  the  same,  to  be  as  any  given  quantity  M 
to  another  given  quantity  N;  then  produce  AB  to  G,  so  that  BG  may  be 
to  CE  as  M  —  N  to  N ;  and  AD  to  H,  so  that  AH  may  be  equal  to  AG  ; 
arid  DF  to  K,  so  that  DK  may  be  to  DH  as  N  to  M.  Join  KB,  and  about 
the  centre  D  with  the  interval  DH  describe  a  circle  meeting  KB  produced 
in  L,  and  draw  BF  parallel  to  DL;  and  the  point  F  will  touch  the  line 
EF,  which,  being  turned  round  the  axis  AB,  will  describe  the  superficies 
sought.  Q.E.F. 

For  conceive  the  lines  CP,  CQ  to  be  every  where  perpendicular  to  AD, 
DF,  and  the  lines  ER,  ES  to  FB,  FD  respectively,  and  therefore  QS  to 
be  always  equal  to  CK;  and  (by  Cor.  2,  Prop.  XCVII)  PD  will  be  to  QD 
as  M  to  N,  and  therefore  as  DL  to  DK,  or  FB  to  FK  ;  and  by  division  as 
DL  —  FB  or  PH  —  PD  —  FB  to  FD  or  FQ  —  QD  ;  arid  by  composition 
as  PH—  FB  to  FQ,  that  is  (because  PH  and  CG,  QS  and  CE,  are  equal), 
as  CE  +  BG  —  FR  to  CE  —  FS.  But  (because  BG  is  to  CE  as  M  — 
N  to  N)  it.  comes  to  pass  also  that  CE  +  BG  is  to  CE  as  M  to  N;  and 
therefore,  by  division,  FR  is  to  FS  as  M  to  N ;  and  therefore  (by  Cor.  2, 
Prop  XCVI1)  the  superficies  EF  compels  a  body,  falling  upon  it  in  the 
direction  DF,  to  go  on  in  the  line  FR  to  the  place  B.  Q.E.D. 

SCHOLIUM. 

,In  the  same  manner  one  may  go  on  to  three  or  more  superficies.  But 
of  all  figures  the  sphserical  is  the  most  proper  for  optical  uses.  If  the  ob 
ject  glasses  of  telescopes  were  made  of  two  glasses  of  a  sphaerical  figure, 
containing  water  between  them,  it  is  not  unlikely  that  the  errors  of  the 
refractions  made  in  the  extreme  parts  of  the  superficies  of  the  glasses  may 
be  accurately  enough  corrected  by  the  refractions  of  the  water.  Such  ob 
ject  glasses  are  to  be  preferred  before  elliptic  and  hyperbolic  glasses,  not  only 
because  they  may  be  formed  with  more  ease  and  accuracy,  but  because  the 
pencils  of  rays  situate  without  the  axis  of  the  glass  would  be  more  accu 
rately  refracted  by  them.  But  the  different  refrangibility  of  different  raya 
is  the  real  obstacle  that  hinders  optics  from  being  made  perfect  by  sphaeri 
cal  or  any  other  figures.  Unless  the  errors  thence  arising  can  be  corrected, 
all  the  labour  spent  in  correcting  the  others  is  quite  thrown  away. 


BOOK    II 


BOOK  II. 


OF  THE  MOTION  OF  BODIES. 

SECTION  I. 
Of  the  motion  of  bodies   that   are  resisted  in  the  ratio  of  the  velocity. 

PROPOSITION  I.    THEOREM  I. 

Tf  a  body  is  resisted  in  the  ratio  of  its  velocity,  the  motion  lost  by  re 
sistance  is  as  the  space  gone  over  in  its  motion. 

For  since  the  motion  lost  in  each  equal  particle  of  time  is  as  the  velocity, 
that  is,  as  the  particle  of  space  gone  over,  then,  by  composition,  the  motion 
lost  in  the  whole  time  will  he  as  the  whole  space  gone  over.  Q.E.D. 

COR.  Therefore  if  the  body,  destitute  of  all  gravity,  move  by  its  innate 
force  only  in  free  spaces,  and  there  be  given  both  its  whole  motion  at  the 
beginning,  and  also  the  motion  remaining  after  some  part  of  the  way  is 
gone  over,  there  will  be  given  also  the  whole  space  which  the  body  can  de 
scribe  in  an  infinite  time.  For  that  space  will  be  to  the  space  now  de 
scribed  as  the  whole  motion  at  the  beginning  is  to  the  part  lost  of  that 
motion. 

LEMMA  I. 

Quantities  proportional  to  their  differences  are  continually  proportional. 
Let  A  be  to  A  —  B  as  B  to  B  —  C  and  C  to  C  —  D,  (fee.,  and,  by  con 
version,  A  will  be  to  B  as  B  to  C  and  C  to  D,  &c.     Q.E.D. 

PROPOSITION  II.    THEOREM  II. 

If  a  body  is  resisted  in  the  ratio  of  its  velocity,  and  moves,  by  its  vis  in- 
sita  only,  through  a  similar  medium,  and  the  times  be  taken  equal, 
the  velocities  in  the  beginning  of  each  of  the  times  are  in  a  geometri 
cal  progression,  and  the  spaces  described  in  each  of  the  times  are  as 
the  velocities. 

CASE  1.  Let  the  time  be  divided  into  equal  particles ;  and  if  at  the  very 
beginning  of  each  particle  we  suppose  the  resistance  to  act  with  one  single 
impulse  which  is  as  the  velocity,  the  decrement  of  the  velocity  in  each  of 


THE    MATHEMATICAL    PRINCIPLES  [BOOK    II. 

the  particles  of  time  will  be  as  the  same  velocity.  Therefore  the  veloci 
ties  are  proportional  to  their  differences,  and  therefore  (by  Lem.  1,  Book 
II)  continually  proportional.  Therefore  if  out  of  an  equal  number  of  par 
ticles  there  be  compounded  any  equal  portions  of  time,  the  velocities  at  the 
beginning  of  those  times  will  be  as  terms  in  a  continued  progression,  which 
are  taken  by  intervals,  omitting  every  where  an  equal  number  of  interme 
diate  terms.  But  the  ratios  of  these  terms  are  compounded  of  the  equaj 
ratios  of  the  intermediate  terms  equally  repeated,  and  therefore  are  equal 
Therefore  the  velocities,  being  proportional  to  those  terms,  are  in  geomet 
rical  progression.  Let  those  equal  particles  of  time  be  diminished,  and 
their  number  increased  in  infinitum,  so  that  the  impulse  of  resistance  may 
become  continual ;  and  the  velocities  at  the  beginnings  of  equal  times,  al 
ways  continually  proportional,  will  be  also  in  this  case  continually  pro 
portional.  Q.E.D. 

CASE  2.  And,  by  division,  the  differences  of  the  velocities,  that  is,  the 
parts  of  the  velocities  lost  in  each  of  the  times,  are  as  the  wholes ;  but  the 
spaces  described  in  each  of  the  times  are  as  the  lost  parts  of  the  velocities 
(by  Prop.  1,  Book  I),  and  therefore  are  also  as  the  wholes.     Q.E.D. 
TT  COROL.  Hence  if  to  the  rectangular  asymptotes  AC,  CH, 

the  hyperbola  BG  is  described,  and  AB,  DG  be  drawn  per 
pendicular  to  the  asymptote  AC,  and  both  the  velocity  of 
.  the  body,  and  the  resistance  of  the  medium,  at  the  very  be 
ginning  of  the  motion,  be  expressed  by  any  given  line  AC, 
and,  after  some  time  is  elapsed,  by  the  indefinite  line  DC ;  the  time  may 
be  expressed  by  the  area  ABGD,  and  the  space  described  in  that  time  by 
the  line  AD.  For  if  that  area,  by  the  motion  of  the  point  D,  be  uniform 
ly  increased  in  the  same  manner  as  the  time,  the  right  line  DC  will  de 
crease  in  a  geometrical  ratio  in  the  same  manner  as  the  velocity ;  and  the 
parts  of  the  right  line  AC,  described  in  equal  times,  will  decrease  in  the 
same  ratio. 

PROPOSITION  III.     PROBLEM  I. 

To  define  the  motion  of  a  body  which,  in  a  similar  medium,  ascends  or 
descends  in  a  right  line,  and  is  resisted  in  the  ratio  of  its  velocity,  and 
acted  upon  by  an  uniform  force  of  gravity. 

The  body  ascending,  let  the  gravity  be  expound 
ed  by  any  given  rectangle  BACH ;  and  the  resist 
ance  of  the  medium,  at  the  beginning  of  the  ascent, 
by  the  rectangle  BADE,  taken  on  the  contrary  side 
Jfl  e  B^l  |  L-  of  the  right  line  AB.  Through  the  point  B,  with 
the  rectangular  asymptotes  AC,  CH,  describe  an 
hyperbola,  cutting  the  perpendiculars  DE,  de,  ID 


SEC.    I.j  OF    NATURAL    PHILOSOPHY.  253 

G,  g  ;  and  the  body  ascending  will  in  the  time  DGgd  describe  the  space 
EG«-e;  in  the  time  DGBA,  the  space  of  the  whole  ascent  EGB ;  in  the 
time  ABK1,  the  space  of  descent  BFK ;  and  in  the  time  IKki  the  space  of 
descent  KFfk;  and  the  velocities  of  the  bodies  (proportional  to  the  re 
sistance  of  the  medium)  in  these  periods  of  time  will  be  ABED,  ABed,  O, 
ABFI,  AB/z  respectively ;  and  the  greatest  velocity  which  the  body  can 
acquire  by  descending  will  be  BACH. 

For  let  the  rectangle  BACH  be  resolved  into  in 
numerable  rectangles  AA',  K/,  Lm,  M//,  *fea,  which 
shall  be  as  the  increments  of  the  velocities  produced 
in  so  many  equal  times;  then  will  0,  AAr,  AL  Am,  An, 
ifec.,  be  as  the  whole  velocities,  and  therefore  (by  suppo 
sition)  as  the  resistances  of  the  medium  in  the  be-  

ginning  of  each  of  the  equal  times.     Make  AC  to  AJLLB 

AK,  or  ABHC  to  AB/vK,  as  the  force  of  gravity  to  the  resistance  in  the 
beginning  of  the  second  time ;  then  from  the  force  of  gravity  subduct  the 
resistances,  and  ABHC,  K/vHC,  L/HC,  MwHC,  (fee.,  will  be  as  the  abso 
lute  forces  with  which  the  body  is  acted  upon  in  the  beginning  of  each  of 
the  times,  and  therefore  (by  Law  I)  as  the  increments  of  the  velocities,  that 
is,  as  the  rectangles  AA-,  K/,  Lm,  M//,  (fee.,  and  therefore  (by  Lem.  1,  Book 
II)  in  a  geometrical  progression.  Therefore,  if  the  right  lines  K£,  LI 
M/TO,  N//,  &c.,  are  produced  so  as  to  meet  the  hyperbola  in  q,  r,  s,  t,  (fee.. 
the  areas  AB^K,  K</rL,  LrsM,  MsJN,  (fee.,  will  be  equal,  and  there 
fore  analogous  to  the  equal  times  and  equal  gravitating  forces.  But  the 
area  AB^K  (by  Corol.  3,  Lem.  VII  and  VIII,  Book  I)  is  to  the  area  Bkq 
as  K^  to  \kq,  or  AC  to  |AK,  that  is,  as  the  force  of  gravity  to  the  resist 
ance  in  the  middle  of  the  first  time.  And  by  the  like  reasoning,  the  areas 
<?KLr,  rLMs,  sMN/,  (fee.,  are  to  the  areas  qklr,  rims,  smnt,  (fee.,  as  the 
gravitating  forces  to  the  resistances  in  the  middle  of  the  second,  third,  fourth 
time,  and  so  on.  Therefore  since  the  equal  areas  BAKq,  </KLr,  rLMs, 
sMN/,  (fee.,  are  analogous  to  the  gravitating  forces,  the  areas  Bkq,  qklr, 
rims,  smut,  (fee.,  will  be  analogous  to  the  resistances  in  the  middle  of 
each  of  the  times,  that  is  (by  supposition),  to  the  velocities,  and  so  to  the 
spaces  described.  Take  the  sums  of  the  analogous  quantities,  and  the  areas 
Bkq,  Elr,  Ems,  But,  (fee.,  will  be  analogous  to  the  whole  spaces  described ; 
and  also  the  areas  AB<?K,  ABrL,  ABsM,  AB^N,  (fee.,  to  the  times.  There 
fore  the  body,  in  descending,  will  in  any  time  ABrL  describe  the  space  Blr, 
and  in  the  time  Lr^N  the  space  rlnt.  Q,.E.D.  And  the  like  demonstra 
tion  holds  in  ascending  motion. 

COROL.  1.  Therefore  the  greatest  velocity  that  the  body  can  acquire  by 
falling  is  to  the  velocity  acquired  in  any  given  time  as  the  £iven  force  ol 
gravity  which  perpetually  acts  upon  it  to  the  resisting  force  which  opposes 
it  at  the  end  of  that  time. 


854 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK    IL 


COROL.  2.  But  the  time  being  augmented  in  an  arithmetical  progression, 
the  sum  of  that  greatest  velocity  and  the  velocity  in  the  ascent,  and  also 
their  difference  in  the  descent,  decreases  in  a  geometrical  progression. 

COROL.  3.  Also  the  differences  of  the  spaces,  which  are  described  in  equal 
differences  of  the  times,  decrease  in  the  same  geometrical  progression. 

COROL.  4.  The  space  described  by  the  body  is  the  difference  of  two 
spaces,  whereof  one  is  as  the  time  taken  from  the  beginning  of  the  descent, 
and  the  other  as  the  velocity;  which  [spaces]  also  at  the  beginning  of  the 
descent  are  equal  among  themselves. 

PROPOSITION  IV.     PROBLEM  II. 

Supposing  the  force  of  gravity  in  any  similar  medium  to  be  uniform, 
and  to  tend  perpendicularly  to  the  plane  of  the  horizon  ;  to  define  the 
motion  of  a  projectile  therein,  which  suffers  resistance  proportional  to 
its  velocity. 

Let  the  projectile  go  from  any  place  D  in 
the  direction  of  any  right  line  DP,  and  let 
its  velocity  at  the  beginning  of  the  motion 
be  expounded  by  the  length  DP.  From  the 
point  P  let  fall  the  perpendicular  PC  on  the 
horizontal  line  DC,  and  cut  DC  in  A,  so 
that  DA  may  be  to  AC  as  the  resistance 
of  the  medium  arising  from  the  motion  up 
wards  at  the  beginning  to  the  force  of  grav 
ity;  or  (which  comes  to  the  same)  so  that 
t  ie  rectangle  under  DA  and  DP  may  be  to 
that  under  AC  and  CP  as  the  whole  resist 
ance  at  the  beginning  of  the  motion  to  the 
force  of  gravity.  With  the  asymptotes 
DC,  CP  describe  any  hyperbola  GTBS  cut 
ting  the  perpendiculars  DG,  AB  in  G  and 
B  ;  complete  the  parallelogram  DGKC,  and 
let  its  side  GK  cut  AB  in  Q,.  Take 


N  in  the  same  ratio  to  QB  as  DC  is  in  to  CP  ;  and  from  any  point  R  of  the 
right  line  DC  erect  RT  perpendicular  to  it,  meeting  the  hy]  erbola'in  T, 
and  the  right  lines  EH,  GK,  DP  in  I,  t,  and  V  ;  in  that  perpendicular 


take  Vr  equal  to  —  ~-  ,  or  which  is  the  same  thing,  take  Rr  equal  to 

(""""PIT? 

—  ^T—  ;  and  the  projectile  in  the  time  DRTG  will  arrive  at  the  point  r 

describing  the  curve  line  DraF,  the  locus  of  the  point  r  ;  thence  it  will 
come  to  its  greatest  height  a  in   the  perpendicular  AB  j  and  afterwards 


SEC.    1.J  »  OF    NATURAL    PHILOSOPHY.  255 

ever  approach  to  the  asymptote  PC.  And  its  velocity  in  any  pjint  r  will 
be  as  the  tangent  rL  to  the  curve.  Q.E.I. 

For  N  is  to  Q,B  as  DC  to  CP  or  DR  to  RV,  and  therefore  RV  is  equal  to 

PR  X  QB      ,'  -•..".    "v       v        DRXQB-/GT 

—  ^r  -  ,  and  R/'  (that  is,  RV  —  Vr,  or  -     --  ^  ---  )  is  equal  to 

DR  X  Ap  —  RDGT 

--  ~  ---  .     JNow  let  the   time   be   expounded    by    the   area 

RDGT  and  (by  Laws,  Cor.  2),  distinguish  the  motion  of  the  body  into 
two  others,  one  of  ascent,  the  other  lateral.  And  since  the  resistance  is  as 
the  motion,  let  that  also  be  distinguished  into  two  parts  proportional  and 
contrary  to  the  parts  of  the  motion  :  and  therefore  the  length  described  by 
the  lateral  motion  will  be  (by  Prop.  II,  Book  II)  as  the  line  DR,  and  the 
height  (by  Prop.  Ill,  Book  II)  as  the  area  DR  X  AB  —  RDGT,  that  is, 
as  the  line  Rr.  But  in  the  very  beginning  of  the  motion  the  area  RDGT 
is  equal  to  the  rectangle  DR  X  AQ,  and  therefore  that  line  Rr  (or 
DRx  AB  — 


that  is,  as  CP  to  DC  ;  and  therefore  as  the  motion  upwards  to  the  motion 
lengthwise  at  the  beginning.  Since,  therefore,  Rr  is  always  as  the  height, 
and  DR  always  as  the  length,  and  Rr  is  to  DR  at  the  beginning  as  the 
height  to  the  length,  it  follows,  that  Rr  is  always  to  DR  as  the  height  to 
the  length  ;  and  therefore  that  the  body  will  move  in  the  line  DraF.  which 
is  the  locus  of  the  point  r.  QJE.D. 

DR  X  AB       RDGT 

COR.  1.  Therefore  Rr  is  equal  to  --  ^  ------  ^  -  .  and  therefore 

if  RT  be  produced  to  X  so  that  RX  may  be  equal  to  --  ^  --  ;  that  is, 

if  the  parallelogram  ACPY  be  completed,  and  DY  cutting  CP  in  Z  be 
drawn,  and  RT  be  produced  till  it  meets  DY  in  X  ;  Xr  will  be  equal  to 

RDGT 

—  ^  —  ,  and  therefore  proportional  to  the  time. 

COR.  2.  Whence  if  innumerable  lines  CR,  or,  which  is  the  same,  innu 
merable  lines  ZX,  be  taken  in  a  geometrical  progression,  there  will  be  as 
many  lines  Xr  in  an  arithmetical  progression.  And  hence  the  curve  DraF 
is  easily  delineated  by  the  table  of  logarithms. 

COR.  3.  If  a  parabola  be  constructed  to  the  vertex  D,  and  the  diameter 
DG  produced  downwards,  and  its  latus  rectum  is  to  2  DP  as  the  whole 
resistance  at  the  beginning  of  the  notion  to  the  gravitating  force,  the  ve 
locity  with  which  the  body  ought  *o  go  from  the  place  D,  in  the  direction 
of  the  right  line  DP,  so  as  in  an  uniform  resisting  medium  to  describe  the 
curve  DraF,  will  be  the  same  as  that  with  which  it  ought  to  go  from  the 
same  place  D  in  the  direction  of  the  same  right  line  DP,  so  as  to  describe 


256 


THE    MATHEMATICAL    PRINCIPLES      ~ 


[BOOK  II 


I    a  parabola  in  a  non-resisting  medium.     For 
the  latus  rectum  of  this  parabola,  at  the  very 

DV2 

beginning  of  the  motion,  is  -y-',  andVris 

tGT      DR  x  T* 

-~JT-  or £^T — .     But  a  right  line,  which, 

if  drawn,  would  touch  the  hyperbola  GTS  in 
K  G,  is  parallel  to  DK,  and  therefore  T*  is 


CKX  DR 


c 


QBx  DC 


^ ,  and  N  is  —  ~pp •  Ahd  there- 


DC 

DR2  X  CK  x  CP 
fore  Vr  is  equal  to       2DC2  X  QlT~;  *^at  *S  (Because  D^  an<*  ^)C,  DV 

DV2  x  CK  ~x  OP 

and  DP  are  proportionals),  to  — ^T5 Fcrr —  J  an<*  tne  ^atus  reeturn 


DV2 


-  comes  out  - 


2DP2  X  QB 


are  proportional), 


CK  X  CP 
2DP2  X  DA 


AC  X  CP 

CP  X  AC ;  that  is,  as  the  resistance  to  the  gravity. 


(because 


and  therefore  is  to  2DP  as  DP  X  DA  to 


Q.E.D. 


COR.  4.  Hence  if  a  body  be  projected  from 
any  place  D  with  a  given  velocity,  in  the 
direction  of  a  right  line  DP  given  by  posi 
tion,  and  the  resistance  of  the  medium,  at 
the  beginning  of  the  motion,  be  given,  the 
curve  DraF,  which  that  body  will  describe, 
may  be  found.  For  the  velocity  being 
given,  the  latus  rectum  of  the  parabola  is 
given,  as  is  well  known.  And  taking  2DP 
to  that  latus  rectum,  as  the  force  of  gravity 
to  the  resisting  force,  DP  is  also  given. 
Then  cutting  DC  in  A,  so  that  GP  X  AC 
may  be  to  DP  X  DA  in  the  same  ratio  of 
the  gravity  to  the  resistance,  the  point  A 
will  be  given.  And  hence  the  curve  DraF 
is  also  given. 

COR.  5.  And,  on  the  contrary,  if  the 
curve  DraF  be  given,  there  will  be  given 
x>th  the  velocity  of  the  body  and  the  resistance  of  the  medium  in  each  of 
the  places  r.  For  the  ratio  of  CP  X  AC  to  DP  X  DA  being  given,  there 
is  given  both  the  resistance  of  the  medium  at  the  beginning  of  the  motion 
and  the  latus  rectum  of  the  parabola ;  and  thence  the  velocity  at  the  be 
ginning  of  the  motion  is  given  also.  Then  from  the  length  of  the  tangent 


SEC.  I.]  OF    NATURAL    PHILOSOPHY.  257 

L  there  is  given  both  the  velocity  proportional  to  it,  and  the  resistance 
proportional  to  the  velocity  in  any  place  r. 

COR.  6.  But  since  the  length  2DP  is  to  the  latus  rectum  of  the  para 
bola  as  the  gravity  to  the  resistance  in  D ;  and,  from  the  velocity  aug 
mented,  the  resistance  is  'ti  gmented  in  the  same  ratio,  but  the  latus  rectum 
of  the  parabola  is  augmented  in  the  duplicate  of  that  ratio,  it  is  plain  thot 
the  length  2DP  is  augmented  in  that  simple  ratio  only  ;  and  is  therefore 
always  proportional  to  the  velocity  ;  nor  will  it  be  augmented  or  dimin 
ished  by  the  change  of  the  angle  CDP,  unless  the  velocity  be  also  changed. 

COR.  7.  Hence  appears  the  method  of  deter 
mining  the  curve  DraF  nearly  from  the  phe- 
nomena,  and  thence  collecting  the  resistance  and 
velocity  with  which  the  body  is  projected.  Let 
two  similar  and  equal  bodies  be  projected  with 
the  same  velocity,  from  the  place  D,  in  differ 
ent  angles  CDP,  CDp  ;  and  let  the  places  F, 
f.  where  they  fall  upon  the  horizontal  plane 
DC,  be  known.  Then  taking  any  length  for  D  */  F 

DP  or  Dp  suppose  the  resistance  in  D  to  be  to 
the  suavity  in  any  ratio  whatsoever,  and  let  that 

ratio  be  expounded  by  any  length  SM.     Then,         , _ 

by  computation,  from  that  assumed  length  DP,  ^x 

find  the  lengths  DF,  D/;  and  from  the  ratio 

F/ 

-p^,  found  by  calculation,  subduct  the  same  ratio  as  found  by  experiment ; 

and  let  the  cKfference  be  expounded  by  the  perpendicular  MN.  Repeat  the 
same  a  second  and  a  third  time,  by  assuming  always  a  new  ratio  SM  of  the 
resistance  to  the  gravity,  and  collecting  a  new  difference  MN.  Draw  the 
affirmative  differences  on  one  side  of  the  right  line  SM,  and  the  negative 
on  the  other  side ;  and  through  the  points  N,  N,  N,  draw  a  regular  curve 
NNN.  cutting  the  right  line  SMMM  in  X,  and  SX  will  be  the  true  ratio 
of  the  resistance  to  the  gravity,  which  was  to  be  found.  From  this  ratio 
the  length  DF  is  to  be  collected  by  calculation  ;  and  a  length,  which  is  to 
the  assumed  length  DP  as  the  length  DF  known  by  experiment  to  the 
length  DF  just  now  found,  will  be  the  true  length  DP.  This  being  known, 
you  will  have  both  the  curve  line  DraF  which  the  body  describes,  and  also 
the  velocity  and  resistance  of  the  body  in  each  place. 

SCHOLIUM. 

But,  yet,  that  the  resistance  of  bodies  is  in  the  ratio  of  the  velocity,  is  more 
a  mathematical  hypothesis  than  a  physical  one.  In  mediums  void  of  all  te 
nacity,  the  resistances  made  to  bodies  are  in  the  duplicate  ratio  of  the  ve 
locities.  For  by  the  action  of  a  swifter  body,  a  greater  motion  in  propor- 

17 


THE    MATHEMATICAL    PRINCIPLES  [BoOK    IL 

tion  to  a  greater  velocity  is  communicated  to  the  same  quantity  of  the 
medium  in  a  less  time  ;  and  in  an  equal  time,  by  reason  of  a  greater  quan 
tity  of  the  disturbed  medium,  a  motion  is  communicated  in  the  duplicate 
ratio  greater ;  and  the  resistance  (by  Law  II  and  III)  is  as  the  motion 
communicated.  Let  us,  therefore,  see  what  motions  arise  from  this  law  of 
resistance. 


SECTION  II. 

'If  the  motion  of  bodies  that  are  resisted  in  tfie  duplicate  ratio  of  their 

velocities. 

PROPOSITION  V.     THEOREM  III. 

Ff  a  body  is  resisted  in  the  duplicate  ratio  of  its  velocity,  and  moves  by 
its  innate  force  only  through  a  similar  medium;  and  the  times  be 
taken  in  a  geometrical  progression,  proceeding  from  less  to  greater 
terms  :  I  say,  that  the  velocities  at  the  beginning  of  each  of  the  times 
are  in  the  same  geometrical  progression  inversely  ;  and  that  the  spaces 
are  equal,  which  are  described  in  each  of  the  times. 
For  since  the  resistance  of  the  medium  is  proportional  to  the  square  of 
the  velocity,  and  the  decrement  of  the  velocity  is  proportional  to  the  resist 
ance  :  if  the  time  be  divided  into  innumerable  equal  particles,  the  squares  of 
the  velocities  at  the  beginning  of  each  of  the  times  will  be  proportional  to 
the  differences  of  the  same  velocities.     Let  those  particles  of  time  be  AK, 
KL,  LM,  &c.,  taken  in  the  right  line  CD;  and 
erect  the  perpendiculars  AB,  Kk,  L/,  Mm,  &c., 
meeting  the  hyperbola  BklmG,  described  with  the 
centre  C,  and  the  rectangular  asymptotes  CD,  CH. 
in  B,  kj  I,  m,  (fee. ;  then  AB  will  be  to  Kk  as  CK 
to  CA,  and,  by  division,  AB  —  Kk  to  Kk  as  AK 

C       ARIMT  to  ^A>  an(1  alternate^  AB —  ^C  to  AK  as  Kk 

to  CA ;  and  therefore  as  AB  X  Kk  to  AB  X  CA. 

Therefore  since  AK  and  AB  X  CA  are  given,*  AB  —  Kk  will  be  as  AB 
X  Kk  ;  and,  lastly,  when  AB  and  KA*  coincide,  as  AB2.  And,  by  the  like 
reasoning,  KAr-U,  J  J-M/??,  (fee.,  will  be  as  Kk2.  LI2,  (fee.  Therefore  the 
squares  of  the  lines  AB,  KA",  L/,  Mm,  (fee.,  are  as  their  differences ;  and, 
therefore,  since  the  squares  of  the  velocities  were  shewn  above  to  be  as  their 
differences,  the  progression  of  both  will  be  alike.  This  being  demonstrated 
it  follows  also  that  the  areas  described  by  these  lines  are  in  a  like  progres 
sion  with  the  spaces  described  by  these  velocities.  Therefore  if  the  velo 
city  at  the  beginning  of  the  first  time  AK  be  expounded  by  the  line  AB, 


SEC.    II.]  OF    NATURAL    PHILOSOPHY.  25CJ 

and  the  velocity  at  the  beginning  of  the  second  time  KL  by  the  line  K& 
and  the  length  described  in  the  hrst  time  by  the  area  AKA*B,  all  the  fol 
lowing  velocities  will  be  expounded  by  the  following  lines  \J.  Mm,  .fee. 
and  the  lengths  described,  by  the  areas  K/,  I  mi.  &c.  And,  by  compo 
sition,  if  the  whole  time  be  expounded  by  AM,  the  sum  of  its  parts,  the 
whole  length  described  will  be  expounded  by  AM/ftB  the  sum  of  its  parts. 
Now  conceive  the  time  AM  to  be  divided  into  the  parts  AK,  KL,  LM,  (fee 
so  that  CA,  CK,  CL,  CM,  (fee.  may  be  in  a  geometrical  progression ;  and 
those  parts  will  be  in  the  same  progression,  and  the  velocities  AB,  K/r, 
L/,  Mm,  (fee.,  will  be  in  the  same  progression  inversely,  and  the  spaces  de 
scribed  Ak,  K/,  Lw,  (fee.,  will  be  equal.  Q,.E.D. 

COR.  1.  Hence  it  appears,  that  if  the  time  be  expounded  by  any  part 
AD  of  the  asymptote,  and  the  velocity  in  the  beginning  of  the  time  by  the 
ordinate  AB,  the  velocity  at  the  end  of  the  time  will  be  expounded  by  the 
ordinate  DG ;  and  the  whole  space  described  by  the  adjacent  hyperbolic 
area  ABGD  ;  and  the  space  which  any  body  can  describe  in  the  same  time 
AD,  with  the  first  velocity  AB,  in  a  non-resisting  medium,  by  the  rectan 
gle  AB  X  AD. 

COR  2.  Hence  the  space  described  in  a  resisting  medium  is  given,  by 
taking  it  to  the  space  described  with  the  uniform  velocity  AB  in  a  non- 
resisting  medium,  as  the  hyperbolic  area  ABGD  to  the  rectangle  AB  X  AD. 

COR.  3.  The  resistance  of  the  medium  is  also  given,  by  making  it  equal, 
in  the  very  beginning  of  the  motion,  to  an  uniform  centripetal  force,  which 
could  generate,  in  a  body  falling  through  a  non-resisting  medium,  the  ve 
locity  AB  in  the  time  AC.  For  if  BT  be  drawn  touching  the  hyperbola 
in  B.  and  meeting  the  asymptote  in  T,  the  right  line  AT  will  be  equal  to 
AC,  and  will  express  the  time  in  which  the  first  resistance,  uniformly  con 
tinned,  may  take  away  the  whole  velocity  AB 

COR.  4.  And  thence  is  also  given  the  proportion  of  this  resistance  to  the 
force  of  gravity,  or  ay  other  given  centripetal  force. 

COR.  5.  And,  vice  versa,  if  there  is  given  the  proportion  of  the  resist- 
;  nee  to  any  given  centripetal  force,  the  time  AC  is  also  given,  in  which  c 
centripetal  force  equal  to  the  resistance  may  generate  any  velocity  as  AB  ; 
and  thence  is  given  the  point  B.  through  which  the  hyperbola,  having  CH 
CD  for  its  asymptotes,  is  to  be  described  :  as  also  the  space  ABGD,  which  a 
body,  by  beginning  its  motion  with  that  velocity  AB,  can  describe  in  any 
time  AD.  in  a  similar  resisting  medium. 

PROPOSITION  VI.     THEOREM  lVrc 

Homogeneous  and  equal  spherical  bodies,  opposed  hy  resistances  that  are 
in  the  duplicate  ratio  of  the  velocities,  and  moving  on  by  their  innate 
force  only,  will,  in  times  which  are  reciprocally  as  the  velocities  at  thr. 


260  THE    MATHEMATICAL    PRINCIPLES  |BOOK    II, 


v 


beg-in  fiing,  describe  equal  spaces,  and  lose  parts  of  their  velocities  pro 
portional  to  the  wholes. 

To  the  rectangular  asymptotes  CD,  CH  de 
scribe  any  hyperbola  B6Ee,  cutting  the  perpen 
diculars  AB,  rib,   DE,  de  in  B,  b,  E,  e;  let  the 
initial  velocities  be  expounded  by  the  perpendicu 
lars  AB,  DE,  and  the  times  by  the  lines  Aa,  Drf. 
Therefore  as  Aa  is  to  l)d,  so  (by  the  hypothesis) 
.  is  DE  to  AB,  and  so  (from  the  nature  of  the  hy- 
C     "^  perbola)  is  CA  to  CD  ;  and,  by  composition,  so  is 

Crt  to  Cd.  Therefore  the  areas  ABba,  DEerf,  that  is,  the  spaces  described, 
are  equal  among  themselves,  and  the  first  velocities  AB,  DE  are  propor 
tional  to  the  last  ab,  de ;  and  therefore,  by  division,  proportional  to  the 
parts  of  the  velocities  lost,  AB  —  ab,  DE  —  de.  Q.E.D. 

PROPOSITION  VII.     THEOREM  V. 

If  spherical  bodies  are  resisted  in  the  duplicate  ratio  of  their  velocities, 
in  times  which  are  as  the  first  motions  directly,  and  the  first  resist 
ances  inversely,  they  will  lose  parts  of  their  motions  proportional  to  the 
wholes,  and  will  describe  spaces  proportional  to  those  times  and  the 
first  velocities  conjunct  It/. 

For  the  parts  of  the  motions  lost  are  as  the  resistances  and  times  con 
junctly.  Therefore,  that  those  parts  may  be  proportional  to  the  wholes, 
the  resistance  and  time  conjunctly  ought  to  be  as  the  motion.  Therefore  the 
time  will  be  as  the  motion  directly  and  the  resistance  inversely.  Where 
fore  the  particles  of  the  times  being  taken  in  that  ratio,  the  bodies  will 
always  loso  parts  of  their  motions  proportional  to  the  wholes,  and  there 
fore  will  retain  velocities  always  proportional  to  their  first  velocities. 
And  because  of  the  given  ratio  of  the  velocities,  they  will  always  describe 
spaces  which  are  as  the  first  velocities  and  the  times  conjunctly.  Q.E.D. 
COR.  1.  Therefore  if  bodies  equally  swift  are  resisted  in  a  duplicate  ra 
tio  of  their  diameters,  homogeneous  globes  moving  with  any  velocities 
whatsoever,  by  describing  spaces  proportional  to  their  diameters,  will  lose 
parts  of  their  motions  proportional  to  the  wholes.  For  the  motion  of  each 
o-lobe  will  be  as  its  velocity  and  mass  conjunctly,  that  is,  as  the  velocity 
and  the  cube  of  its  diameter ;  the  resistance  (by  supposition)  will  be  as  the 
square  of  the  diameter  and  the  square  of  the  velocity  conjunctly ;  and  the 
•time  (by  this  proposition)  is  in  the  former  ratio  directly,  and  in  the  latter 
inversely,  that  is,  as  the  diameter  directly  and  the  velocity  inversely ;  and 
therefore' the  space,  which  is  proportional  to  the  time  and  velocity  is  as 
the  diameter. 

COR.  2.  If  bodies  equally  swift  are  resisted  in  a  sesquiplicate  ratio  of 
their  diameters,  homogeneous  globes,  moving  with  any  velocities  whatso- 


SEC.    1L]  OF    NATURAL    PHILOSOPHY.  261 

ever,  by  describing  spaces  that  are  in  a  sesquiplicate  ratio  of  the  diameters, 
will  lose  parts  of  their  motions  proportional  to  the  wholes. 

COR.  3.  And  universally,  if  equally  swift  bodies  are  resisted  in  the  ratio 
of  any  power  of  the  diameters,  the  spaces,  in  which  homogeneous  globes, 
moving  with  any  velocity  whatsoever,  will  lose  parts  of  their  motions  pro 
portional  to  the  wholes,  will  be  as  the  cubes  of  the  diameters  applied  to 
that  power.  Let  those  diameters  be  D  and  E  :  and  if  the  resistances,  where 
the  velocities  are  supposed  equal,  are  as  T)n  and  E" ;  the  spaces  in  which 
the  globes,  moving  with  any  velocities  whatsoever,  will  lose  parts  of  their 
motions  proportional  to  the  wholes,  will  be  as  D3  — n  and  E3  —  n.  And 
therefore  homogeneous  globes,  in  describing  spaces  proportional  to  D 3  —  n 
and  E3  — n,  will  retain  their  velocities  in  the  same  ratio  to  one  another  as 
at  the  beginning. 

COR.  4.  Now  if  the  globes  are  not  homogeneous,  the  space  described  by 
the  denser  globe  must  be  augmented  in  the  ratio  of  the  density.  For  the 
motion,  with  an  equal  velocity,  is  greater  in  the  ratio  of  the  density,  and 
the  time  (by  this  Prop.)  is  augmented  in  the  ratio  of  motion  directly,  and 
the  space  described  in  the  ratio  of  the  time. 

COR.  5.  And  if  the  globes  move  in  different  mediums,  the  space,  in  a 
medium  which,  cccteris  paribus,  resists  the  most,  must  be  diminished  in  the 
ratio  of  the  greater  resistance.  For  the  time  (by  this  Prop.)  will  be  di 
minished  in  the  ratio  of  the  augmented  resistance,  and  the  space  in  the  ra 
tio  of  the  time. 

LEMMA  II. 

The  moment  of  any  genitum  is  equal  to  the  moments  of  each  of  the  gen- 
eratinrr  sides  drawn  into  the  indices  of  the  powers  of  those  sides,  and 
into  their  co-efficients  continually. 

I  call  any  quantity  a  genitum  which  is  not  made  by  addition  or  sub- 
duction  of  divers  parts,  but  is  generated  or  produced  in  arithmetic  by  the 
multiplication,  division,  or  extraction  of  the  root  of  any  terms  whatsoever : 
in  geometry  by  the  invention  of  contents  and  sides,  or  of  the  extremes  and 
means  of  proportionals.  Quantities  of  this  kind  are  products,  quotients, 
roots,  rectangles,  squares,  cubes,  square  and  cubic  sides,  and  the  like. 
These  quantities  I  here  consider  as  variable  and  indetermined,  and  increas 
ing  or  decreasing,  as  it  were,  by  a  perpetual  motion  or  flux ;  and  I  under 
stand  their  momentaneous  increments  or  decrements  by  the  name  of  mo 
ments  ;  so  that  the  increments  may  be  esteemed  as  added  or  affirmative 
moments ;  and  the  decrements  as  subducted  or  negative  ones.  But  take 
care  not  to  look  upon  finite  particles  as  such.  Finite  particles  are  not 
moments,  but  the  very  quantities  generated  by  the  moments.  We  are  to 
conceive  them  as  the  just  nascent  principles  of  finite  magnitudes.  Nor  do 
we  in  this  Lemma  regard  the  magnitude  of  the  moments,  but  their  firsf 


262  THE    MATHEMATICAL    PRINCIPLES  [BoOK    11 

proportion,  as  nascent.  It  will  be  the  same  thing,  if,  instead  of  moments, 
we  use  either  the  velocities  of  the  increments  and  decrements  (which  may 
also  be  called  the  motions,  mutations,  and  fluxions  of  quantities),  or  any 
finite  quantities  proportional  to  those  velocities.  The  co-efficient  of  any 
generating  side  is  the  quantity  which  arises  by  applying  the  genitum  to 
ihat  side. 

Wherefore  the  sense  of  the  Lemma  is,  that  if  the  moments  of  any  quan 
tities  A,  B,  C,  &c.,  increasing  or  decreasing  by  a  perpetual  flux,  or  the 
velocities  of  the  mutations  which  are  proportional  to  them,  be  called  a,  6, 
r,  (fee.,  the  moment  or  mutation  of  the  generated  rectangle  AB  will  be  «B 
-h  b A ;  the  moment  of  the  generated  content  ABC  will  be  aBC  -f  b AC  4 

-1         -2.         .1 

cAB;  and  the  moments  of  the  generated  powers  A2.  A3,  A4,  A2,  A2.  A3, 
A*,  A  —  ',  A  —  2,  A—  *  will  be  2aA,  3a  A2,  4aA3,  |«A  —  *,  f«A* 

11  3 

i«  A —  s,  |/iA  —  3,  —  a  A  —  2,  — 2aA —  3,  —  £aA —  2  respectively;  and 
in  general,  that  the  moment  of  any  power  A^,  will  be  ^  aAn-^.  Also, 
that  the  moment  of  the  generated  quantity  A2B  will  be  2aAB  4-  bA~  ;  the 
moment  of  the  generated  quantity  A 3B4C2  will  be  3«A2B4C2  +  4/>A3 

A3 
B3C2  4-2cA3B'C;  and  the  moment  of  the  generated  quantity  —   or 

A  »B  — 2  will  be  3aA 2 B  —  2  —  2bA 3B  —  3 ;  and  so  on.  The  Lemma  is 
thus  demonstrated. 

CASE  1.  Any  rectangle,  as  AB,  augmented  by  a  perpetual  flux,  when,  as 
yet,  there  wanted  of  the  sides  A  and  B  half  their  moments  \a  and  \b,  was 
A  —  \a  into  B  —  \b,  or  AB  —  ±a  B  —  \b  A  +  \ab  ;  but  as  soon  as  the 
sides  A  and  B  are  augmented  by  the  other  half  moments,  the  rectangle  be 
comes  A  4-  4-a  into  B  4-  \b,  or  AB  -f  ^a  B  4-  \b  A  -f  \ab.  From  this 
rectangle  subduct  the  former  rectangle,  and  there  will  remain  the  exces.? 
aE  -f  bA.  Therefore  with  the  whole  increments  a  and  b  of  the  sides,  tin 
increment  aB  +  f>A  of  the  rectangle  is  generated.  Q.K.D. 

CASE  2.  Suppose  AB  always  equal  to  G,  and  then  the  moment  of  the 
content  ABC  or  GC  (by  Case  1)  will  be^C  +  cG,  that  is  (putting  AB  and 
aB  +  bA  for  G  and  «*),  aBC  -h  bAC  4-  cAB.  And  the  reasoning  is  the 
same  for  contents  under  ever  so  many  sides.  Q.E.D. 

CASE  3.  Suppose  the  sides  A,  B,  and  C,  to  be  always  equal  among  them 
selves;  and  the  moment  «B  +  />A,  of  A2,  that  is,  of  the  rectangle  AB, 
will  be  2aA  ;  and  the  moment  aBC  +  bAC  +  cAB  of  A3,  that  is,  of  the 
content  ABC,  will  be  3aA2.  And  by  the  same  reasoning  the  moment  of 
any  power  An  is  naAn — '.  Q.E.D 

CASE  4.  Therefore  since  -7  into  A  is  1,  the  moment  of  -r-  drawn  into 
A  A 


SEC.    11.]  OF    NATURAL    PHILOSOPHY.  263 

A,  together  with  —  drawn  into  a.  will  be  the  moment  of  1,  that  is,  nothing. 
A 

Therefore  the  moment  of  -r,  or  of  A —  ',  is  -r— .     And  generally  since 

A  .A 

T-  into  An  is  I,  the  moment  of  —drawn  into  An  together   with  —    into 
A n  A.  A n 

naA"  —  !  will  be  nothing.     And,  therefore,  the  moment  of  -r-  or  A  — n 

A 

will   be—  £T^~7-     Q-E.D. 

V  .  t  .  '  i 

CASE  5.  And  since  A2  into  A2  is  A,  the  moment  of  A1  drawn  into  2 A2 

will  be  a  (by  Case  3) ;  and,  therefore,  the  moment  of  A7  will  be  n~r~r  or 

^A-j 

£#A  —  £.     And,  generally,  putting  A~^  equal  to  B,  then  Am  will  be  equal 
to  Bn,  and  therefore  maAm  —  !  equal  to  nbBn  —  ' ,  and  ma  A  —  '  equal  to 

?tbB  —  ',  or  tib  A  —  ^  5  an<i  therefore  ri  a  A  —^~  is  equal  to  &,  that  is,  equal 

to  the  moment  of  A^.     Q.E.D. 

CASE  6.  Therefore  the  moment  of  any  generated  quantity  AmBn  is  the 
moment  of  Am  drawn  into  Bn,  together  with  the  moment  of  Bn  drawn  into 
A",  that  is,  maAm —  '  B"  -f-  nbBn —  !  Am;  and  that  whether  the  indices 
in  arid  n  of  the  powers  be  whole  numbers  or  fractions,  affirmative  or  neg 
ative.  And  the  reasoning  is  the  same  for  contents  under  more  powers. 
Q.E.D. 

COR.  1.  Henoe  in  quantities  continually  proportional,  if  one  term  is 
given,  the  moments  of  the  rest  of  the  terms  will  be  as  the  same  terms  mul 
tiplied  by  the  number  of  intervals  between  them  and  the  given  term.  Let 
A,  B,  C,  D;  E,  F,  be  continually  proportional ;  then  if  the  term  C  is  given, 
the  moments  of  the  rest  of  the  terms  will  be  among  themselves  as  —  2A, 
—  B?  D,  2E,  3F. 

COR.  2.  And  if  in  four  proportionals  the  two  means  are  given,  the  mo 
ments  of  the  extremes  will  be  as  those  extremes.  The  same  is  to  be  un 
derstood  of  the  sides  of  any  given  rectangle. 

COR.  3.  And  if  the  sum  or  difference  of  two  squares  is  given,  the  mo 
ments  of  the  sides  will  be  reciprocally  as  the  sides. 

SCHOLIUM. 

In  a  letter  of  mine  to  Mr.  /.  Collins,  dated  December  10,  1672,  having 
described  a  method  of  tangents,  which  I  suspected  to  be  the  same  with 
Slusius*s  method,  which  at  that  time  wag  not  made  public,  I  subjoined  these 
words  •  This  is  one  particular,  or  rather  a  Corollary,  of  a  general  nte 


264  THE    MATHEMATICAL    PRINCIPLES  [BjOK    II. 

thod,  which  extends  itself,  without  any  troublesome  calculation,  not  ojdy 
to  the  drawing  of  tangents  to  any  curve  lines,  whether  geometrical  or 
mechanical,  or  any  how  respecting  right  lines  or  other  cnrves,  but  also 
to  the  resolving  other  abstrnser  kinds  of  problems  about  the  crookedness, 
areas,  lengths,  centres  of  gravity  of  curves,  &c. ;  nor  is  it  (as  Hudd^ri's 
method  de  Maximis  &  Minimia)  limited  to  equations  which  are  free  from 
surd  quantities.  This  method  I  have  interwoven  with  that  other  oj 
working  in  equations,  by  reducing  them  to  infinite  serie?.  So  far  that 
letter.  And  these  last  words  relate  to  a  treatise  I  composed  on  that  sub 
ject  in  the  year  1671.  The  foundation  of  that  general  method  is  contain 
ed  in  the  preceding  Lemma. 

PROPOSITION  VIII.     THEOREM  VI. 

If  a  body  in  an  uniform  medium,  being  uniformly  acted  upon  by  the  force 
of  gravity,  ascends  or  descends  in  a  right  line  ;  and  the  whole  space 
described  be  distinguished  into  equal  parts,  and  in  the  beginning  of 
each  of  the  parts  (by  adding  or  subducting  the  resisting  force  of  the 
medium  to  or  from  the  force  of  gravity,  when  the  body  ascends  or  de 
scends]  yon  collect  the  absolute  forces  ;  I  say,  that  those  absolute  forces 
ire  in  a  geometrical  progression. 

For  let  the  force  of  gravity  be  expounded  by  the 
given  line  AC  ;  the  force  of  resistance  by  the  indefi 
nite  line  AK  ;  the  absolute  force  in  the  descent  of  the 
body  by  the  difference  KC  :  the  velocity  of  the  I  tody 
<^LKJL&i>F/ by  a  line  AP,  which  shall  be  a  mean  proportional  be 
tween  AK  and  AC,  and  therefore  in  a  subduplicate  ratio  of  the  resistance; 
the  increment  of  the  resistance  made  in  a  given  particle  of  time  by  the  li- 
neola  KL,  and  the  contemporaneous  increment  of  the  velocity  by  the  li- 
neola  PQ  ;  and  with  the  centre  C,  and  rectangular  asymptotes  CA,  CH, 
describe  any  hyperbola  BNS  meeting  the  erected  perpendiculars  AB,  KN, 
LO  in  B,  N  and  O.  Because  AK  is  as  AP2 ,  the  moment  KL  of  the  one  will 
be  as  the  moment  2APQ  of  the  other,  that  is,  as  AP  X  KC  ;  for  the  in 
crement  PQ  of  the  velocity  is  (by  Law  II)  proportional  to  the  generating 
force  KC.  Let  the  ratio  of  KL  be  compounded  with  the  ratio  KN,  and 
the  rectangle  KL  X  KN  will  become  as  AP  X  KC  X  KN  ;  that  is  (because 
the  rectangle  KC  X  KN  is  given),  as  AP.  But  the  ultimate  ratio  of  the 
hyperbolic  area  KNOL  to  the  rectangle  KL  X  KN  becomes,  when  the 
points  K  and  L  coincide,  the  ratio  of  equality.  Therefore  that  hyperbolic 
evanescent  area  is  as  AP.  Therefore  the  whole  hyperbolic  area  ABOL 
is  composed  of  particles  KNOL  which  are  always  proportional  to  the 
velocity  AP;  and  therefore  is  itself  proportional  to  the  space  described 
with  that  velocity.  Let  ,that  area  be  now  divided  into  equal  parts 


SEC.  IJ.J 


OF    NATURAL    PHILOSOPHY. 


265 


as  ABMI,  IMNK,  KNOL,  (fee.,  and  the  absolute  forces  AC,  1C,  KC,  LC, 

(fee.,  will  be  in  a  geometrical  progression.  Q,.E.D.  And  by  a  like  rea 
soning,  in  the  ascent  of  the  body,  taking,  on  the  contrary  side  of  the  point 
A,  the  equal  areas  AB?m,  i/nnk,  knol,  (fee.,  it  will  appear  that  the  absolute 
forces  AC.  iG,  kC,  1C,  (fee.,  are  continually  proportional.  Therefore  if  all 
the  spaces  in  the  ascent  and  descent  are  taken  equal,  all  the  absolute  forces 
1C,  kC,  iC,  AC,  1C,  KC,  LC,  (fee.,  will  be  continually  proportional.  Q,.E.D. 

COR.  1.  Hence  if  the  space  described  be  expounded  by  the  hyperbolic 
area  ABNK,  the  force  of  gravity,  the  velocity  of  the  body,  and  the  resist 
ance  of  the  medium,  may  be  expounded  by  the  lines  AC,  AP,  and  AK  re 
spectively  •  and  vice  versa. 

COR.  2.  And  the  greatest  velocity  which  the  body  can  ever  acquire  in 
an  infinite  descent  will  be  expounded  by  the  line  AC. 

COR.  3.  Therefore  if  the  resistance  of  the  medium  answering  to  any 
given  velocity  be  known,  the  greatest  velocity  will  be  found,  by  taking  it 
to  that  given  velocity  in  a  ratio  subduplicate  of  the  ratio  which  the  force 
of  gravity  bears  to  that  known  resistance  of  the  medium. 

PROPOSITION  IX.     THEOREM  VII. 

Supposing  ivhat  is  above  demonstrated,  I  say,  that  if  the  tangents  of  t-he 
angles  of  the  sector  of  a  circle,  and  of  an  hyperbola,  be  taken  propor 
tional  to  the  velocities,  the  radius  being  of  a  fit  magnitude,  all  the  time 
of  the  ascent  to  the  highest  place  icill  be  as  the  sector  of  the  circle,  and 
all  the  time  of  descending  from  the  highest  place  as  the  sector  of  t/ie 
hyperbola. 

To   the   right   line   AC,  which   ex 
presses  the  force  of  gravity,  let  AD 
drawn  perpendicular  and  equal.     From 
the   centre    D  with  the  semi-diameter 
AD  describe  as  well  the  cmadrant  A^E 

-t 

of  a  circle,  as  the  rectangular  hyper 
bola  AVZ,  whose  axis  is  AK,  principal 
vertex  A,  and  asymptote  DC.  Let  Dp, 
DP  be  drawn ;  and  the  circular  sector 
AtD  will  be  as  all  the  time  of  the  as 
cent  to  the  highest  place ;  and  the  hy 
perbolic  sector  ATD  as  all  the  time  of  descent  from  the  highest  place;  ii 
BO  be  that  the  tangents  Ap,  AP  of  those  sectors  be  as  the  velocities. 

CASE  1.  Draw  Dvq  cutting  off  the  moments  or  least  particles  tDv  and 
^  ?,  described  in  the  same  time,  of  the  sector  ADt  and  of  the  triangle 
AD/?.  Since  those  particles  (because  of  the  common  angle  D)  are  in  a  du 
plicate  ratio  of  the  sides,  the  particle  tDv  will  be  as  — -^-^-— ,  that  is 


266 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK    li. 


(because  tD  is  given),  as  ^f.  But  joD8  is  AD3  +  Ap2,  that  is,  AD2  -h 
AD  X  AA-,  or  AD  X  Gk ;  and  (/Dp  is  1  AD  X  pq.  Therefore  tDv,  the 

BO 

particle  of  the  sector,  is  as  ^ ,  ;  that  is,  as  the  least  decrement  pq  of  the 

velocity  directly,  and  the  force  Gk  which  diminishes  the  velocity,  inversely ; 
and  therefore  as  the  particle  of  time  answering  to  the  decrement  of  the  ve 
locity.  And,  by  composition,  the  sum  of  all  the  particles  tDv  in  the  sector 
AD/  will  be  as  the  sum  of  the  particles  of  time  answering  to  each  of  the 
lost  particles  pq  of  the  decreasing  velocity  Ap,  till  that  velocity,  being  di 
minished  into  nothing,  vanishes;  that  is,  the  whole  sector  AD/  is  as  the 
whole  time  of  ascent  to  the  highest  place.  Q.E.D. 

CASE  2.  Draw  DQV  cutting  off  the  least  particles  TDV  and  PDQ  of 
the  sector  DAV,  and  of  the  triangle  DAQ ;  and  these  particles  will  be  to 
each  other  as  DT2  to  DP2,  that  is  (if  TX  and  AP  are  parallel),  as  DX2 
to  DA2  or  TX2  to  AP2  ;  and,  by  division,  as  DX2  —  TX2  to  DA2  - 
AP2.  But.  from  the  nature  of  the  hyperbola,  DX2  — TX2  is  AD2  ;  and,  by 
the  supposition,  AP2  is  AD  X  AK.  Therefore  the  particles  are  to  each 
other  as  AD2  to  AD2  —  AD  X  AK  ;  that  is,  as  AD  to  AD  —  AK  or  AC 

;  and 


to  CK  :  and  therefore  the  particle  TDV  of  the  sector  is  - 

PQ 


therefore  (because  AC  and  AD  are  given)  as 


CK 

that  is,  as  the  increment 


of  the  velocity  directly,  and  as  the  force  generating  the  increment  inverse 
ly  ;  and  therefore  as  the  particle  of  the  time  answering  to  the  increment. 
And,  by  composition,  the  sum  of  the  particles  of  time,  in  which  all  the  par 
ticles  PQ  of  the  velocity  A  I3  are  generated,  will  be  as  the  sum  of  the  par 
ticles  of  the  sector  ATI)  ;  that  is,  the  whole  time  will  be  as  the  whole 
sector.  Q.E.D. 

COR.  1.  Hence  if  AB  be  equal  to  a 
fourth  part  of  AC,  the  space  which  a  body 
will  describe  by  falling  in  any  time  will 
be  to  the  space  which  the  body  could  de 
scribe,  by  moving  uniform]}'  on  in  the 
same  time  with  its  greatest  velocity 
AC,  as  the  area  ABNK,  which  ex 
presses  the  space  described  in  falling  to 
the  area  ATD,  which  expresses  the 
time.  For  since  AC  is  to  AP  as  AP 
_  to  AK,  then  (by  Cor.  1,  Lem.  II,  of  this 

Book)  LK  is  to  PQ  as  2AK  to  AP,  that  is,  as  2AP  to  AC,  and  thence 
LK  is  to  ^PQ  as  AP  to  JAC  or  AB  ;  and  KN  is  to  AC  or  AD  as  AB  tc 


.  II.]  OF    NATURAL    PHILOSOPHY.  267 

UK ;  and  therefore,  ex  cequo,  LKNO  to  DPQ,  as  AP  to  CK.  But  DPQ 
was  to  DTV  as  CK  to  AC.  Therefore,  ex  aquo,  LKNO  is  to  DTV  r,? 
AP  to  AC ;  that  is,  as  the  velocity  of  the  falling  body  to  the  greatest 
velocity  which  the  body  by  falling  can  acquire.  Since,  therefore,  the 
moments  LKNO  and  DTV  of  the  areas  ABNK  and  ATD  are  as  the  ve 
locities,  all  the  parts  of  those  areas  generated  in  the  same  time  will  be  as 
the  spaces  described  in  the  same  time ;  and  therefore  the  whole  areas  ABNK 
and  ADT,  generated  from  the  beginning,  will  be  as  the  whole  spaces  de 
scribed  from  the  beginning  of  the  descent.  Q.E.D. 

COR.  2.  The  same  is  true  also  of  the  space  described  in  the  ascent. 
That  is  to  say,  that  all  that  space  is  to  the  space  described  in  the  same 
time,  with  the  uniform  velocity  AC,  as  the  area  ABttk  is  to  the  sector  ADt. 

COR.  3.  The  velocity  of  the  body,  falling  in  the  time  ATD,  is  to  the 
velocity  which  it  would  acquire  in  the  same  time  in  a  non-resisting  space, 
as  the  triangle  APD  to  the  hyperbolic  sector  ATD.  For  the  velocity  in 
a  non-resisting  medium  Avould  be  as  the  time  ATD,  and  in  a  resisting  me 
dium  is  as  AP,  that  is,  as  the  triangle  APD.  And  those  velocities,  at  the 
beginning  of  the  descent,  are  equal  among  themselves,  as  well  as  those 
areas  ATD,  APD. 

COR.  4.  By  the  same  argument,  the  velocity  in  the  ascent  is  to  the  ve 
locity  with  which  the  body  in  the  same  time,  in  a  non-resisting  space,  would 
lose  all  its  motion  of  ascent,  as  the  triangle  ApD  to  the  circular  sector 
AtD  ;  or  as  the  right  line  Ap  to  the  arc  At. 

COR.  5.  Therefore  the  time  in  which  a  body,  by  falling  in  a  resisting 
medium,  would  acquire  the  velocity  AP,  is  to  the  time  in  which  it  would 
acquire  its  greatest  velocity  AC,  by  falling  in  a  non-resisting  space,  as  the 
sector  ADT  to  the  triangle  ADC :  and  the  time  in  which  it  would  lose  its 
velocity  Ap,  by  ascending  in  a  resisting  medium,  is  to  the  time  in  which 
it  would  lose  the  same  velocity  by  ascending  in  a  non-resisting  space,  as 
the  arc  At  to  its  tangent  Ap. 

COR.  6.  Hence  from  the  given  time  there  is  given  the  space  described  in 
the  ascent  or  descent.  For  the  greatest  velocity  of  a  body  descending  in 
wfinitum  is  given  (by  Corol.  2  and  3,  Theor.  VI,  of  this  Book) ;  and  thence 
the  time  is  given  in  which  a  body  would  acquire  that  velocity  by  falling 
in  a  non-resisting  space.  And  taking  the  sector  ADT  or  ADt  to  the  tri 
angle  ADC  in  the  ratio  of  the  given  time  to  the  time  just  now  found, 
there  will  be  given  both  the  velocity  AP  or  Ap,  and  the  area  ABNK  or 
AB//&,  which  is  to  the  sector  ADT,  or  AD/,  as  the  space  sought  to  the 
space  which  would,  in  the  given  time,  be  uniformly  described  with  that 
greatest  velocity  found  just  before. 

COR.  7.  And  by  going  backward,  from  the  given  space  of  ascent  or  de 
scent  AB»?A:  or  ABNK,  there  will  be  given  the  time  AD*  or  ADT. 


268 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK    ii 


PROPOSITION  X.     PROBLEM  III. 

Suppose  the  uniform  force  of  gravity  to  tend  directly  to  the  plane  of  the 
horizon,  and  the  resistance  to  be  as  the  density  of  the  medium  and  the 
square  of  the  velocity  coiijunctly  :  it  is  proposed  to  find  the  density  of 
the  medium  in  each  place,  which  shall  make  the  body  move  in  any 
given  carve  line  ;  the  velocity  of  the  body  and  the  resistance  of  the 
medium  in  each  place. 

Let  PQ  be  a  plane  perpendicular  to 
the  plane  of  the  scheme  itself;  PFHQ 
a  curve  line  meeting  that  plane  in  the 
points  P  and  Q ;  G,  H,  I,  K  four 
places  of  the  body  going  on  in  this 
\  curve  from  F  to  Q ;  and  GB,  HC,  ID, 
KE  four  parallel  ordinates  let  fall 

p  A.      33    c^D  E   Q  from  these  points  to  the  horizon,  and 

standing  on  the  horizontal  line  PQ  at  the  points  B,  C,  D,  E ;  and  let  the 
distances  BC,  CD,  DE,  of  the  ordinates  be  equal  among  themselves.  From 
the  points  G  and  H  let  the  right  lines  GL,  HN,  be  drawn  touching  the 
curve  in  G  and  H,  and  meeting  the  ordinates  CH,  DI,  produced  upwards, 
in  L  and  N  :  and  complete  the  parallelogram  HCDM.  And  the  times  in 
which  the  body  describes  the-  arcs  GH,  HI,  will  be  in  a  subduplicate  ratio 
of  the  altitudes  LH,  NI;  which  the  bodies  would  describe  in  those  times, 
by  falling  from  the  tangents;  and  the  velocities  will  be  as  the  lengths  de 
scribed  GH,  HI  directly,  and  the  times  inversely.  Let  the  times  be  ex- 

C*TT          TTT 
pounded  by  T  and  t,  and  the  velocities  by  —•=-  and  ---;  and  the  decrement 

J_  L 

/-^TT  TTT 

of  the  velocity  produced  in  the  time  t  will  be  expounded  by  -7^ — . 

This  decrement  arises  from  the  resistance  which  retards  the  body,  and  from 
the  gravity  which  accelerates  it.  Gravity,  in  a  falling  body,  which  in  its 
fall  describes  the  space  NI,  produces  a  velocity  with  which  it  would  be  able 
to  describe  twice  that  space  in  the  same  time,  as  Galileo  has  demonstrated  ; 

2NI 


that  is,  the  velocity 


:  but  if  the  body  describes  the  arc  HI,  it  augments 
MIxNl 


HI 


— ;   and  therefore  generates 


that  arc  only  by  the  length  HI  —  HN  or 

only  the  velocity  — iff—-     Let  this  velocity  be  added  to  the  before- 

t  X   H.JL 

mentioned  decrement,  and  we  shall  have  the  decrement  of  the  velocity 

GH         HI       SMI  X  Nl 

arising  from   the  resistance  alone,   that  is,   -^ : — h 


T 


SEC.  II.J 


OF    NATURAL    PHILOSOPHY. 


269 


2NI. 


Therefore  since,  in  the  same  time,  the  action  of  gravity  generates,  in  a  fall 
ing  body,  the  velocity  ,  the  resistance  will  be  to  the  gravity  as  7^ 

HI         2MI X  NI      2NI  t  X  GH  2MI  X  NI 

—  +    — TTT-  to  —  or  as  — ^ — —  HI  -f 

Now  for  the  abscissas  CB,  CD, 
CE,  put  —  o,  o,  2o.  For  the  ordinate 
CH  put  P ;  and  for  MI  put  any  series 
Qo  +  Ro2  +  So3  +,  &c.  And  all 
the  terms  of  the  series  after  the  first, 
that  is,  Ro2  +  So3  +,  (fee.,  will  be 
NI ;  and  the  ordinates  DI,  EK,  and 

BGwill  be  P  —  Qo—Ro2— So3— p  A      B   c  T>  E 

(fee.,  P  —  2Qo—  4Ro2—  SSo3— ,  (fee.,  and  P  -\-  Qo  — Ro2  +  So3—, 
(fee.,  respectively.  And  by  squaring  the  differences  of  the  ordinates  BG  — 
CH  and  CH  —  DI,  and  to  the  squares  thence  produced  adding  the  squares 
of  BC  and  CD  themselves,  you  will  have  oo -f-  QQoo  —  2QRo3  +,  (fee., 
and  oo  -f  QQoo  -f  2QRo3  +,  (fee.,  the  squares  of  the  arcs  GH,  HI ;  whose 

QRoo  QRoo 

roots  o  y  -  • ,  and  o  </!  4-  QQ  4-      — are  the 

1  +  QQ       v/1  +  QQ  s/1  -f  QQ 

arcs  GH  and  HI.  Moreover,  if  from  the  ordinate  CH  there  be  subducted 
half  the  sum  of  the  ordinates  BG  and  DI,  and  from  the  ordinate  DI  there 
be  subducted  half  the  sum  of  the  ordinates  CH  and  EK,  there  will  remain 
Roo  and  Roo  +  3So3,  the  versed  sines  of  the  arcs  GI  and  HK.  And  these 
are  proportional  to  the  lineolae  LH  and  NI,  and  therefore  in  the  duplicate 

ratio  of  the  infinitely  small  times  T  and  t :  and  thence  the  ratio  ~,  is   ^ 


R  +  SSo      R  -f 

^ or 


So         ,  t  X  GH       TTT      2MI  X  NI    , 

—  :  and  — ^ HI  H TTT ,  by  substituting 


R  T  HI 

the  values  of  ™,    GH,    HI,   MI   and   NI  just  found,   becomes   -^- 

J-  /w-Lt/ 


I  +  QQ.     Arid  since  2NI  is  2Roo,  the  resistance  will  be  now  to  the 

OO 

gravity  as  --     TT'Q     to  2Roo>  that  is>  as  3S      r  to  4RR. 


And  the  velocity  will  be  such,  that  a  body  going  off  therewith  from  any 
place  H,  in  the  direction  of  the  tangent  HN,  would  describe,  in  vacuo,  a 

parabola,  whose  diameter  is  HC,  and  its  latus  rectum    NT    or  --  ^  ----  . 

And  the  resistance  is  as  the  density  of  the  medium  and  the  square  of 
the  velocity  conjunctly  ;  and  therefore  the  density  of  the  medium  is  as  the 
resistance  directly,  and  the  square  of  the  velocity  inversely  ;  that  is,  as 


270  THE    MATHEMATICAL    PRINCIPLES  [BOOK  II. 


QQ __ 


4RR 

Q.E.I. 

COR.  1.  If  the  tangent  HN  be  produced  both  ways,  so  as  to  meet  any 

HT 
ordinatc  AF  in  T  -       will  be  equal  to  V/T+  QQ;  and  therefore  in  what 


has  gone  before  may  be  put  for  ^  \  -\-  QQ.  By  this  means  the  resistance 
will  be  to  the  gravity  as  3S  X  HT  to  4RR  X  AC ;  the  velocity  will  be  a* 

•r-pj — --^,  and  the  density  of  the  medium  will  be  as  „—  -TT-n. 
AO  -v/  ±i  Jti  X  H 1 

COR.  2.  And  hence,  if  the  curve  line  PFHQ  be  denned  by  the  relation 
between  the  base  or  abscissa  AC  and  the  ordinate  CH,  as  is  usual,  and  the 
value  of  the  ordinate  be  resolved  into  a  converging  series,  the  Problem 
will  be  expeditiously  solved  by  the  first  terms  of  the  series ;  as  in  the  fol 
lowing  examples. 

EXAMPLE  1.  Let  the  line  PFHQ,  be  a  semi-circle  described  upon  the 
diameter  PQ,  to  find  the  density  of  the  medium  that  shall  make  a  projec 
tile  move  in  that  line. 

Bisect  the  diameter  PQ  in  A  ;  and  call  AQ,  n  ;  AC,  a  ;  CH,  e  ;  and 
CD,  o  ;  then  DI2  or  AQ,2  —  AD 2  =  nn  —  aa  —  2ao  —  oo,  or  ec.  —  2ao 
—  oo  ;  and  the  root  being  extracted  by  our  method,  will  give  DI  =  e  — 
ao       oo        aaoo        ao3        a3  o3 
~e~~~2e 2e?  ~~~  W  ~2? '  &C*     Here  put  nn  f°r  ee  +  aa>  and 

ao         nnoo       anno3 

DI  will  become  =  e — ,  &c. 

e          2e3          2e5 

Such  series  I  distinguish  into  successive  terms  after  this  manner :  I  call 
that  the  first  term  in  which  the  infinitely  small  quantity  o  is  not  found ; 
the  second,  in  which  that  quantity  is  of  one  dimension  only ;  the  third,  in 
which  it  arises  to  two  dimensions ;  the  fourth,  in  which  it  is  of  three ;  and 
so  ad  infinitum.  And  the  first  term,  which  here  is  e,  will  always  denote 
the  length  of  the  ordinate  CH,  standing  at  the  beginning  of  the  indefinite 

quantity  o.     The  second  term,  which  here  is  — ,  will  denote  the  difference 

between  CH  and  DN ;  that  is,  the  lineola  MN  which  is  cut  off  by  com 
pleting  the  parallelogram  HC  DM;  and  therefore  always  determines  the 

CM? 

position  of  the  tangent  HN ;  as,  in  this  case,  by  taking  MN  to  HM  as  — 

G 

to  o,  or  a  to  e.  The  third  term,  which  here  is  -£—,  will  represent  the  li 
neola  IN,  which  lies  between  the  tangent  and  the  curve ;  and  therefore 
determines  the  angle  of  contact  IHN,  or  the  curvature  which  the  curve  line 


SEC.    II.]  OF    NATURAL    PHILOSOPHY.  271 

has  in  H.  If  that  lineola  IN  is  of  a  finite  magnitude,  it  will  be  expressed 
by  the  third  term,  together  with  those  that  follow  in  wfinitu:.:i.  But  if 
that  lineola  be  diminished  in.  infini- 
tnm,  the  terms  following  become  in 
finitely  less  than  the  third  term,  and 
therefore  may  be  neglected.  The 
fourth  term  determines  the  variation 
of  the  curvature ;  the  fifth,  the  varia 
tion  of  the  variation  ;  and  so  on. 

Whence,  by  the  way,  appears  no  con-p~"  ~K     B~C~D~E~ 

temptible  use  of  these  series  in  the  solution  of  problems  that  depend  upon 
tangents,  and  the  curvature  of  curves. 

ao         77/700         anno 3 

Now  compare  the  series  e  — ^— ^~ &c.,  with  the 

e  Ze3  Ze* 

series  P  —  Qo  —  -  Roo  —  So3  —  &c.,  and  for  P,  Q,  II  and  S?  put  e,  -,  ^-^ 

and   ~ — ,  and  for  ^  1  +  QQ  put        1  H or  -  ;  and  the   density  oi 

2e 5 '  ee        e ' 

the  medium  will  come  out  as  — ;  that  is  (because  n  is  given),  as   -   or 

lie '  e 

~Yj,  that  is,  as  that  length  of  the  tangent  HT,  which  is  terminated  at  the 
OH. 

semi-diameter  AF  standing  perpendicularly  on  PQ :  and  the  resistance 
will  be  to  the  gravity  as  3a  to  2>/,  that  is,  as  SAC  to  the  diameter  PQ  of 
the  circle;  and  the  velocity  will  be  as  i/  CH.  Therefore  if  the  body  goes 
from  the  place  F,  with  a  due  velocity,  in  the  direction  of  a  line  parallel  to 
PQ,  and  the  density  of  the  medium  in  each  of  the  places  H  is  as  the  length 
of  the  tangent  HT,  and  the  resistance  also  in  any  place  H  is  to  the  force 
of  gravity  as  SAC  to  PQ,  that  body  will  describe  the  quadrant  FHQ  of  a 
circle.  Q.E.I. 

But  if  the  same  body  should  go-frorn  the  place  P,  in  the  direction  of  a 
line  perpendicular  to  PQ,  and  should  begin  to  move  in  an  arc  of  the  semi 
circle  PFQ,  we  must  take  AC  or  a  on  the  contrary  side  of  the  centre  A  ; 
and  therefore  its  sign  must  be  changed,  and  we  must  put — a  for  +  a. 

Then  the  density  of  the  medium  would  come  out  as .      But  nature 

does  not  admit  of  a  negative  density,  that  is,  a  density  which  accelerates 
the  motion  of  bodies;  and  therefore  it  cannot  naturally  come  to  pass  that 
a  body  by  ascending  from  P  should  describe  the  quadrant  PF  of  a  circle. 
To  produce  such  an  effect,  a  body  ought  to  be  accelerated  by  an  impelling 
medium,  and  not  impeded  by  a  resisting  one. 

EXAMPLE  2.  Let  the  line  PFQ  be  a  parabola,  having  its  axis  AF  per- 


272 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK  IL 


pendicular  to  the  horizon  PQ,  to  find  the  density  of  the  medium,  which 
will  make  a  projectile  move  in  that  line. 

From  the  nature  of  the  parabola,  the  rectangle  PDQ, 
is  equal  to  the  rectangle  under  the  ordinate  DI  and  some 
given  right  line  ;  that  is,  if  that  right  line  be  called  b  ; 
PC,  a;  PQ,  c;  CH,  e;  and  CD,  o;  the  rectangle  a 
A.  CD    ~Q  +  o  into  c  —  a  —  o  or  ac  —  aa  —  2ao  -{-co  —  oo,  ia 

ac  —  aa 

equal  to  the  rectangle  b  into  DI,  and  therefore  DI  is  equal  to  --  7  --  h 

c  —  2a  oo  c  —  2a 

—  -.  —  o  —  —r.    Now  the  second  term  —  -,  —  o  of  this  series  is  to  be  put 
b  b  b 

oo 

for  Q,o,  and  the  third  term  -r  for  Roo.      But  since  there  are  no   more 

terms,  the  co-efficient  S  of  the  fourth  term  will  vanish  ;  and  therefore  the 

S 
ouantitv   -  —  ,  to  which  the  density  of  the  medium  is  proper- 

' 


R  v  i 

tional,  will  be  nothing.     Therefore,  where  the  medium  is  of  no  density, 

the  projectile  will  move  in  a  parabola  ;  as  Galileo  hath  heretofore  demon 

strated.     Q.E.I. 

EXAMPLE  3.  Let  the  line  AGK  be  an  hyperbola,  having  its  asymptote 

NX  perpendicular  to  the  horizontal  plane  AK,  to  find  the  density  of  the 

medium  that  will  make  a  projectile  move  in  that  line. 

Let  MX  be  the  other  asymptote,  meeting 
the  ordinate  DG  produced  in  V  ;  and  from 
the  nature  of  the  hyperbola,  the  rectangle  of 
XV  into  VG  will  be  given.  There  is  also 
given  the  ratio  of  DN  to  VX,  and  therefore 
the  rectangle  of  DN  into  VG  is  given.  Let 
that  be  bb  :  and,  completing  the  parallelo 
gram  DNXZ,  let  BN  be  called  a;  BD,  o; 
NX,  c;  and  let  the  given  ratio  of  VZ  to 

ZX  or  DN  be  -.     Then  DN  will  be  equal 


MA.  BD  K    N 

bb 
to  a  —  o}  VG  equal  to 


,  VZ  equal  to  —  X  a  —  o.  and  GD  or  NX 
a  —  o'  n 


m 


m 


-VZ— VG  equal  to  c— —  a  +  —  o  — .     Let  the  term  -  —   be 

n  n  a — o  a  —  o 

bb      bb          bb  bb  , 

resolved  into  the  converging  series  ~^"  +  ^°  +  ^l00  +  ^4    °>  &c'»  and 

m          bb      m         bb          bb  bb 

GD  will  become  equal  to  c  —  -  a—  -  +  -o  —  ~  o  — ^  o2  —  51  ° 


SEC.  II.] 


OF    NATURAL    PHILOSOPHY. 


273 


&c.     The  second  term  —  o o  of  this  series  is  to  be  u?ed  for  Qo;  the 

n          aa 

third  —  o2,  with  its  sign  changed  for  Ro2  ;  and  the  fourth  —  o3,  with  its 

m       bb    bb         bb 

sign  changed  also  for  So3,  and  their  coefficients ,  —  and  —  are  to 

n       aa   a  a 

be  pat  for  Q,  R,  and  S  in  the  former  rule.     Which  being  done,  the  den- 

bb 
a* 


sity  of  the  medium  will  come  out  as    , , 


bb 


a 


mm 
nn 


2mbb 
naa 


I 


mm 


-— ,  that  is,  if  in  VZ  you  take  VY  equal  to 


aa 


aa 


1  m2 

VG,  as  YT7-     For  aa  and  —^  a 2 


2mbb        b' 
nn  n  aa 

2mbb       b 4 

— •  H are  the  squares  of  XZ 

n          aa 

and  ZY.  But  the  ratio  of  the  resistance  to  gravity  is  found  to  be  that  of 
3XY  to  2YG ;  and  the  velocity  is  that  with  which  the  body  would  de- 

XY2 

scribe  a  parabola,  whose  vertex  is  G,  diameter  DG,  latus  rectum  ^v  .  Sup 
pose,  therefore,  that  the  densities  of  the  medium  in  each  of  the  places  G 
are  reciprocally  as  the  distances  XY,  and  that  the  resistance  in  any  place 
G  is  to  the  gravity  as  3XY  to  2YG  ;  and  a  body  let  go  from  the  place  A, 
with  a  due  velocity,  will  describe  that  hyperbola  AGK.  Q.E.I. 

EXAMPLE  4.  Suppose,  mdeMtely,  the  line  AGK  to  be  an  hyperbola 
described  with  the  centre  X,  and  the  asymptotes  MX,  NX,  so  that,  having 
constructed  the  rectangle  XZDN,  whose  side  ZD  cuts  the  hyperbola  in  G 
and  its  asymptote  in  V,  VG  may  be  reciprocally  as  any  power  DNn  of  the 
line  ZX  or  DN,  whose  index  is  the  number  n :  to  find  the  density  of  the 
medium  in  which  a  projected  body  will  describe  this  curve. 

For  BN,  BD,  NX,  put  A,  O,  C,  respec-  ^ 

tively,  and  let  VZ  be  to  XZ  or  DN  as  d  to 
bb 


e,  and  VG  be  equal  to 


be  equal  to  A  —  O,  VG  ==  ^= 


then   DN  will 


VZ  = 


O,  and  GD  or  NX  —  VZ  —  VG  equal 


274 

term 


THE    MATHEMATICAL    PRINCIPLES  [BOOK    H 

nbb 


nn  -f-  n 


bb  U  !        J    •     x  •     £     •  *° 

•=rr  be  resolved  into  an  infinite  series  -r-  +  — 
A  —  Of  A"      A.n 

3  -±-  3nn  +  2/i 


X    O    + 


n 


"  ~  x  bb  O3,&c,,andGD  will  be  equal 


g^TT-T  X  00  O2  + 

c£              bb         d                  nbb                 +  ?m  - 
toC  —  -A--T-+-O-  -r O  -  ~ 

e  A"         e  A"  +  l  2An  -f 

+  H  i^T  't"\  bb°3>  &c-     The  second  tcrm  -  O  -  -T 

6An  +  e  An  4-  l 

series  is  to  be  used  for  0,0,  the  third  ^  0  66O2    for  Roo,  the  fourth 

— -£\~r~3 bbO5  for  So3.     And  thence  the  density  of  the  medium 


Oof  this 


-,  in  any  place  G7  will  be 


2dnbb 


nub*' 


and  therefore  if  in  VZ  you  take  VY  equal  to  n  X  VG,  that  density  is  re- 

n  w       IT-  j   ^  *  2rf//66  /mfi4 

ciprocally  as  XY.      For  A2   and   —  A2  --  —  A  +    —  r    are  the 

tc/  o^x  ./\_  " 

squares  of  XZ  and  ZY.     Hut  the  resistance  in  the  same  place  G  is  to  the 
force  of  gravity  as  3S  X       -  to  4RR,  that  is,  as  XY  to 


And  the  velocity  there  is  the  same  wherewith  the  projected  body  would 
move  in  a  parabola,  whose  vertex  is  G,  diameter  GD,  and  latus  rectum 

2XY2 

or  -  --------  -  -  .     Q.E.I. 


R 


nn 


VG 


AC 
HT 


SCHOLIUM. 

In  the  same  manner  that  the  den 
sity  of  the  medium  comes  out  to  be  as 

S  X  AC    . 

« Tjr™  m  ^°r-  1)  if  the  resistance 

lx  X   HI 

is  put  as  any  power  V"  of  the  velocity 
V,  the  density  of  the   medium   will 


come    out      to     be      as 


B    C   D  E    Q 


— .  x 


S 


And  therefore  if  a  curve  can  be  found,  such  that  the  ratio  of to 

4  —  o 

R— i- 


SEC.  II.J 


OF    NATURAL    PHILOSOPHY, 


275 


n  — 1 


,  or  ofgr^  to 


may  be  given ;  the  body,  in  an  uni- 


z 


HT 

AC 

form  medium,  whose  resistance  is  as  the  power  V"  of  the  velocity  V,  will 
move  in  this  curve.  But  let  us  return  to  more  simple  curves. 

Because  there  can  be  no  motion  in  a  para 
bola  except  in  a  non-resisting  medium,  but 
in  the  hyperbolas  here  described  it  is  produced 
by  a  perpetual  resistance ;  it  is  evident  that 
the  line  which  a  projectile  describes  in  an 
uniformly  resisting  medium  approaches  nearer 
to  these  hyperbolas  than  to  a  parabola.  That 
line  is  certainly  of  the  hyperbolic  kind,  but 
about  the  vertex  it  is  more  distant  from  the 
asymptotes,  and  in  the  parts  remote  from  the 

vertex  draws  nearer  to  them  than  these  hy-  M   JL  BD  K 

perbolas  here  described.  The  difference,  however,  is  not  so  great  between 
the  one  and  the  other  but  that  these  latter  may  be  commod^ously  enough 
used  in  practice  instead  of  the  former.  And  perhaps  these  may  prove  more 
useful  than  an  hyperbola  that  is  more  accurate,  and  at  the  same  time  more 
compounded.  They  may  be  made  use  of,  then,  in  this  manner. 

Complete  the  parallelogram  XYGT,  and  the  right  line  GT  will  touch 
the  hyperbola  in  G,  and  therefore  the  density  of  the  medium  in  G  is  re- 

GT2 

ciprocally  as  the  tangent  GT,  and  the  velocity  there  as  ^  -^=-  ;  and  the 


resistance  is  to  the  force  of  gravity  as  GT  to 

Therefore  if  a  body  projected  from  the 
place  A,  in  the  direction  of  the  right  line 
AH,  describes  the  hyperbola  AGK  and 
AH  produced  meets  the  asymptote  NX  in 
H,  arid  AI  drawrri  parallel  to  it  meets  the 
other  asymptote  MX  in  I ;  the  density  of 
the  mediu.n  in  A  will  be  reciprocally  as 
AH.  and  the  velocity  of  the  body  as  -J 

AH1 

.  .  .  and  the  resis'an^e  there  to  the  force 
Al 


2nn 


n  +2 


^-  X  GV. 


.  TT       2nn  +  2n 

of  gravity  rs  AH  to ZiTo  ~ 


X    AI.     Her,ce  the   following    rules  a  e 


deduced. 

RULE  1.  If  the  density  of  the  medium  at  A,  and  the  velocity  with  which 
the  body  is  projected  remain  the  same,  and  the  angle  NAH  be  changed , 
the  lengths  AH,  AI,  HX  will  remain.  Therefore  if  those  lengths,  in  any 


276  THE    MATHEMATICAL    PRINCIPLES  [BOOK    II. 

one  case,  are  found,  the  hyperbola  may  afterwards  be  easily  determined 
from  any  given  angle  NAH. 

RULE  2.  If  the  angle  NAH,  and  the  density  of  the  medium  at  A,  re 
main  the  same,  and  the  velocity  with  which  the  body  is  projected  be 
changed,  the  length  AH  will  continue  the  same  ;  and  AI  will  be  changed 
in  a  duplicate  ratio  of  the  velocity  reciprocally. 

RULE  3.  If  the  angle  NAH,  the  velocity  of  the  body  at  A,  and  the  ac- 
celerative  gravity  remain  the  same,  and  the  proportion  of  the  resistance  at 
A  to  the  motive  gravity  be  augmented  in  any  ratio  ;  the  proportion  of  AH 
to  A I  will  be  augmented  in  the  same  ratio,  the  latus  rectum  of  the  above- 

AH2 

mentioned  parabola  remaining  the  same,  and  also  the  length  propor- 

AI 

tional  to  it ;  and  therefore  AH  will  be  diminished  in  the  same  ratio,  and 
AI  will  be  diminished  in  the  duplicate  of  that  ratio.  But  the  proportion 
of  the  resistance  to  the  weight  is  augmented,  when  either  the  specific  grav 
ity  is  made  less,  the  magnitude  remaining  equal,  or  when  the  density  of 
the  medium  is  made  greater,  or  when,  by  diminishing  the  magnitude,  the 
resistance  becomes  diminished  in  a  less  ratio  than  the  weight. 

RULE  4.  Because  the  density  of  the  medium  is  greater  near  the  vertex 
of  the  hyperbola  than  it  is  in  the  place  A,  that  a  mean  density  may  be 
preserved,  the  ratio  of  the  least  of  the  tangents  GT  to  the  tangent  AH 
ought  to  be  found,  and  the  density  in  A  augmented  in  a  ratio  a  little 
greater  than  that  of  half  the  sum  of  those  tangents  to  the  least  of  the 
tangents  GT. 

RULE  5.  If  the  lengths  AH,  AI  are  given,  and  the  figure  AGK  is  to  be 
described,  produce  HN  to  X,  so  that  HX  may  be  to  AI  as  n  -\-  1  to  1  ;  and 
with  the  centre  X,  and  the  asymptotes  MX,  NX,  describe  an  hyperbola 
through  the  point  A,  such  that  AI  may  be  to  any  of  the  lines  VG  as  XV" 

to  xr. 

RULE  6.  By  how  much  the  greater  the  number  n  is,  so  much  the  more 
accurate  are  these  hyperbolas  in  the  ascent  of  the  body  from  A,  and  less 
accurate  in  its  descent  to  K ;  and  the  contrary.  The  conic  hyperbola 
keeps  a  mean  ratio  between  these,  and  is  more  simple  than  the  rest.  There 
fore  if  the  hyperbola  be  of  this  kind,  and  you  are  to  find  the  point  K, 
where  the  projected  body  falls  upon  any  right  line  AN  passing  through 
the  point  A,  let  AN  produced  meet  the  asymptotes  MX,  NX  in  M  and  N, 
and  take  NK  equal  to  AM. 

RULE  7.  And  hence  appears  an  expeditious  method  of  determining  this 
hyperbola  from  the  phenomena.  Let  two  similar  and  equal  bodies  be  pro 
jected  with  the  same  velocity,  in  different  angles  HAK,  h A k,  and  let  them 
fall  upon  the  plane  of  the  horizon  in  K  and  k  ;  and  note  the  proportion 
of  AK  to  A  A".  Let  it  be  as  d  to  e.  Then  erecting  a  perpendicular  A I  of 
uny  length,  assume  any  how  the  length  AH  or  Ah,  and  thence  graphically, 


SEC.  II. 


OF    NATURAL    PHILOSOPHY. 


27? 


or  by  scale  and  compass,  collect  the  lengths  AK,  A/>*  (by  Rule  6).     If  the 
ratio  of  AK  to  A/.*  bo  the  same  with  that  of  d  to  e,  the  length  of  AH  was 


rightly  assumed.  If  not,  take  on  the  indefinite  right  line  SM,  the  length 
SM  equal  to  the  assumed  AH ;  and  erect  a  perpendicular  MN  equal  to  the 

AK      d 

difference  -r-r of  the  ratios  drawn  into  any  given  right  line.     By  the 

like  method,  from  several  assumed  lengths  AH,  you  may  find  several  points 
N  ;  and  draw  througli  them  all  a  regular  curve  NNXN,  cutting  tr.e  right 
line  SMMM  in  X.  Lastly,  assume  AH  equal  to  the  abscissa  SX,  and 
thence  find  again  the  length  AK ;  and  the  lengths,  w'hich  are  to  the  as 
sumed  length  AI,  and  this  last  AH,  as  the  length  AK  known  by  experi 
ment,  to  the  length  AK  last  found,  will  be  the  true  lengths  AI  and  AH, 
which  were  to  be  found.  But  these  being  given,  there  will  be  given  also 
the  resisting  force  of  the  medium  in  the  place  A,  it  being  to  the  force  of 
gravity  as  AH  to  JAI.  Let  the  density  of  the  medium  be  increased  by 
Rule  4,  and  if  the  resisting  force  just  found  be  increased  in  the  same  ratio, 
it  will  become  still  more  accurate. 

RULE  8.  The  lengths  AH,  HX  being  found  ;  let  there  be  now  re 
quired  the  position  of  the  line  AH,  according  to  which  a  projectile  thrown 
with  that  given  velocity  shall  fall  upon  any  point  K.  At  the  joints  A 
and  K,  erect  the  lines  AC,  KF  perpendicular  to  the  horizon  :  whereof  let 
AC  be  drawn  downwards,  and  be  equal  to  AI  or  ^HX.  With  the  asymp 
totes  AK,  KF,  describe  an  hyperbola,  whose  conjugate  shall  pass  through 
the  point  C  ;  and  from  the  centre  A,  with  the  interval  AH.  describe  a  cir 
cle  cutting  that  hyperbola  in  the  point  H ;  then  the  projectile  thrown  in 
the  direction  of  the  right  line  AH  will  fall  upon  the  point  K.  Q.E.I.  For 
the  point  H,  because  of  the  given  length  AH,  must  be  somewhere  in  the 
circumference  of  the  described  circle.  Draw  CH  meeting  AK  and  KF  in 
E  and  F:  and  because  CH,  MX  are  parallel,  and  AC,  AI  equal,  AE  will 
be  equal  to  AM,  and  therefore  also  equal  to  KN.  But  CE  is  to  AE  as 
FH  to  KN.  and  therefore  CE  and  FH  are  equal.  Therefore  the  point  H 
falls  upon  the  hyperbolic  curve  described  with  the  asymptotes  AK,.KF 
whose  conjugate  passes  through  the  point  C  ;  and  is  therefore  found  in  the 


27S 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK  11 


common  intersection  of  this  hyperbolic 
curve  and  the  circumference  of  the  de 
scribed  circle.  Q.E.D.  It  is  to  be  ob 
served  that  this  operation  is  the  same, 
whether  the  right  line  AKN  be  parallel  to 
the  horizon,  or  inclined  thereto  in  any  an 
gle  :  and  that  from  two  intersections  H, 
//.,  there  arise  two  angles  NAH,  NAA  ; 
and  that  in  mechanical  practice  it  is  suf 
ficient  once  to  describe  a  circle,  then  to 
apply  a  ruler  CH,  of  an  indeterminate  length,  HO  to  the  point  C,  that  its 
part  PH,  intercepted  between  the  circle  and  the  right  line  FK,  may  bo 
equal  to  its  part  CE  placed  between  the  point  C  and  the  right  line  AK 

What  has  been  said  of  hyperbolas  may  be  easily 
applied  to  pir  i'>;>l.i3.  For  if  a  parabola  be  re 
presented  by  XAGK,  touched  by  a  right  line  XV 
in  the  vertex  X,  and  the  ordinates  IA,  YG  be  as 
any  powers  XI",  XV";  of  the  abscissas  XI,  XV  ; 
draw  XT,  GT,  AH,  whereof  let  XT  be  parallel 
to  VG,  and  let  GT,  AH  touch  the  parabola  in 
G  and  A  :  and  a  body  projected  from  any  place 
A,  in  the  direction  of  the  right  line  AH,  with  a 
due  velocity,  will  describe  this  parabola,  if  the  density  of  the  medium  in 
each  of  the  places  G  be  reciprocally  as  the  tangent  GT.  In  that  case  the 
velocity  in  G  will  be  the  same  as  would  cause  a  body,  moving  in  a  non- 
resisting  space,  to  describe  a  conic  parabola,  having  G  for  its  vertex,  VG 

2GT2 

produced  downwards  for  its  diameter,  and -. —  for    its     latus 

nn  —  n  X  VG 

rectum.     And  the  resisting  force  in  G  will  be  to  the  force  of  gravity  as  GT  to 

2nti  —  2tt 

~2~  VG.     Therefore  if  NAK  represent  an  horizontal  line,  and  botli 

the  density  of  the  medium  at  A,  and  the  velocity  with  which  the  body  is 
projected,  remaining  the  same,  the  angle  NAH  be  any  how  altered,  the 
lengths  AH,  AI,  HX  will  remain;  and  thence  will  be  given  the  vertex  X 
of  the  parabola,  and  the  position  of  the  right  line  XI ;  and  by  taking  VG 
to  IA  as  XVn  to  XI",  there  will  be  given  all  the  points  G  of  the  parabola, 
through  which  the  projectile  will  pass. 


SEC.    IILJ  OF    NATURAL    PHILOSOPHY.  279 

SECTION  III. 

Of  the  motions  of  bodies  which  are  resisted  partly  in  the  ratio  of  the  ve 
locities,  and  partly  in  the  duplicate  of  the  same  ratio. 

PROPOSITION  XI.     THEOREM  VIII. 

If  a  body  be  resisted  partly  in  the  ratio  and  partly  in  the  duplicate  ratio 
of  its  velocity,  and  moves  in  a  similar  medium  by  its  innate  force 
only;  and  the  times  be  taken  in  arithmetical  progression;  then 
quantities  reciprocally  proportional  to  the  velocities,  increased  by  a  cer 
tain  given  quantity,  will  be  in  geometrical  progression. 
With  the  centre  C,  and  the  rectangular  asymptotes  ^ 

OADd  and  CH,  describe  an  hyperbola  BEe,  and  let  |  \p 

AB,  DE,  de.  be  parallel  to  the  asymptote  CH.     In   | 

the  asymptote  CD  let  A,  G  be  given  points  ;  and  if 

the  time  be  expounded  by  the  hyperbolic  area  ABED 

uniformly  increasing,  I  say,  that  the  velocity  may  ~r 

be  expressed  by  the  length  DF,  whose  reciprocal 

GD,  together  with  the  given  line  CG,  compose  the 

length  CD  increasing  in  a  geometrical  progression. 

For  let  the  areola  DEec/  be  the  least  given  increment  of  the  time,  and 

Dd  will  be  reciprocally  as  DE,  and  therefore  directly  as  CD.     Therefore 

the  decrement  of  ^TR,  which  (by  Lem.  II,  Book  II)  is  ^no,  will  be  also  as 


D 
tf 


CD       CG  +  GD  1  CG 

GO*  °r  —  GD2  ~  '       fc  1S>aS  GD  +  GJD2*      *nerefore  tne  timc 
uniformly  increasing  by  the  addition  of  the  given  particles  EDcfe,  it  fol 

lows  that  r       decreases  in  the  same  ratio  with  the  velocity.     For  the  de 


crement  of  the  velocity  is  as  the  resistance,  that  is  (by  the  supposition),  as 
the  sum  of  two  quantities,  whereof  one  is  as  the  velocity,  and  the  other  as 

the  square  of  the  velocity  ;  and  the  decrement  of   ~~  is  as  the  sum  of  the 


1  C^(^  1 

quantities  ~-^=r   and  pfp,>  whereof  the  first  is   ^^r   itself,   and   the    last 


i  i 

is  a*  /-TFT;  •  therefore  T^-R  is  as  tne  velocity,  the  decrements  of  both 

-'  CilJ 


being  analogous.     And  if  the  quantity  GD    reciprocally  proportional  to 
T,  be  augmented  by  the  given  quantity  CG  ;  the  sum   CD,  the  time 


ABED  uniformly  increasing,  will  increase  !n  a  geometrical  progression. 
Q.E.D. 


THE    MATHEMATICAL    PRINCIPLES  [BOOK    II 

COR.  1.  Therefore,  if,  having  the  points  A  and  G  given,  the  time  bo 
expounded  by  the  hyperbolic  area  ABED,  the  velocity  may  be  expounded 

by     -r  the  reciprocal  of  GD. 


COR.  2.  And  by  taking  GA  to  GD  as  the  reciprocal  of  the  velocity  at 
the  beginning  to  the  reciprocal  of  the  velocity  at  the  end  of  any  time 
ABED,  the  point  G  will  be  found.  And  that  point  being  found  the  ve 
locity  may  be  found  from  any  other  time  given. 

PROPOSITION  XII.     THEOREM  IX. 

The  same  things  being  supposed,  I  say,  that  if  the  spaces  described  are 
taken  in  arithmetical  progression,  the  velocities  augmented  by  a  cer 
tain  given  quantity  will  be  in  geometrical  progression. 

In  the  asymptote  CD  let  there  be  given  the 
point  R,  and,  erecting  the  perpendicular  RS 
meeting  the  hyperbola  in  S,  let  the  space  de 
scribed  be  expounded  by  the  hyperbolic  area 
I    RSED  ;  and  the  velocity  will  be  as  the  length 

J    GD,  which,  together  with  the  given  line  CG, 

**  composes  a  length  CD  decreasing  in  a  geo 
metrical  progression,  while  the  space  RSED  increases  in  an  arithmetical 
[(regression. 

For,  because  the  incre  nent  EDde  of  the  space  is  given,  the  lineola  DC?, 
which  is  the  decrement  of  GD,  will  be  reciprocally  as  ED,  and  therefore 
directly  as  CD  ;  that  is,  as  the  sum  of  the  same  GD  and  the  given  length 
CG.  But  the  decrement  of  the  velocity,  in  a  time  reciprocally  propor 
tional  thereto,  in  which  the  given  particle  of  space  D^/eE  is  described,  is 
as  the  resistance  and  the  time  conjunctly,  that  is.  directly  as  the  sum  of 
two  quantities,  whereof  one  is  as  the  velocity,  the  other  as  the  square  of 
the  velocity,  and  inversely  as  the  veh  city  ;  and  therefore  directly  as  the 
sum  of  two  quantities,  one  of  which  is  given,  the  other  is-  as  the  velocity. 
Therefore  the  decrement  both  of  the  velocity  and  the  line  GD  is  as  a  given 
quantity  and  a  decreasing  quantity  conjunctly;  and,  because  the  decre 
ments  are  analogous,  the  decreasing  quantities  will  always  be  analogous ; 
viz.,  the  velocity,  and  the  line  GD.  U.E.D. 

COR.  1.  If  the  velocity  be  expounded  by  the  length  GD,  the  space  de 
scribed  will  be  as  the  hyperbolic  area  DESR. 

COR.  2.  And  if  the  point  „  be  assumed  any  how,  the  point  G  will  be 
found,  by  taking  GR  to  GD  as  the  velocity  at  the  beginning  to  the  velo 
city  after  any  space  RSED  is  described.  The  point  G  being  given,  the 
space  is  given  from  the  given  velocity :  and  the  contrary. 

Cotw  3.  Whence  since  (by  Prop.  XI)  the  velocity  is  given  from  the  given 


SEC.  Ilt.1 


Or    NATURAL    PHILOSOPHY. 


281 


time,  and  (by  this  Prop.)  the  space  is  given  from  the  given  velocity ;  the 
space  will  be  given  from  the  given  time :  and  the  contrary. 

PROPOSITION  XKI.     THEOREM  X. 

Supposing  that  a  body  attracted  downwards  by  an  uniform  gravity  as 
cends  or  descends  in  a  right  line;  and  that  the  same  is  resisted 
partly  in  the  ratio  of  its  velocity,  and  partly  in  the  duplicate  ratio 
thereof:  I  say,  that,  if  right  lines  parallel  to  the  diameters  of  a  circle 
and  an  hyperbola,  be  drawn  through  the  ends  of  the,  conjugate  diame 
ters,  and  the  velocities  be  as  some  segments  of  those  parallels  drawn 
from  a  given  point,  the  times  will  be  as  the  sectors  of  the,  areas  cut 
off  by  right  lines  drawn  from  the  centre  to  the  ends  of  the  segments  ; 
and  the  contrary. 

CASE  1.  Suppose  first  that  the  body  is  ascending, 
and  from  the  centre  D,  with  any  semi-diameter  DB, 
describe  a  quadrant  BETF  of  a  circle,  and  through 
the  end  B  of  the  semi-diameter  DB  draw  the  indefi 
nite  line  BAP,  parallel  to  the  semi-diameter  DF.     In 
chat  line  let  there  be  given  the  point  A,  and  take  the 
segment  AP  proportional  to  the  velocity.     And  since 
one  part  of  the  resistance  is  as  the   velocity,  and 
another  part  as  the  square  of  the  velocity,  let  the 
whole  resistance  be  as  AP2  -f  2BAP.     Join  DA,  DP,  cutting  the  circle 
in  E  and  T,  and  let  the  gravity  be  expounded  by  DA2,  so  that  the  gravity 
shall  be  to  the  resistance  in  P  as  DA2  to  AP2+2BAP ;  and  the  time  of  the 
whole  ascent  will  be  as  the  sector  EDT  of  the  circle. 

For  draw  DVQ,,  cutting  off  the  moment  PQ,  of  the  velocity  AP,  and  the 
moment  DTV  of  the  sector  DET  answering  to  a  given  moment  of  time  ; 
and  that  decrement  PQ,  of  the  velocity  will  be  as  the  sum  of  the  forces  of 
gravity  DA2  and  of  resistance  AP2  +  2BAP,  that  is  (by  Prop.  XII 
BookII,Elem.),asDP*.  Then  the  arsa  DPQ,  which  is  proportional  to  PQ: 
is  as  DP2,  and  the  area  DTV,  which  is  to  the  area  DPQ,  as  DT2  to  DP2,  it 
as  the  given  quantity  DT2.  Therefore  the  area  EDT  decreases  uniformly 
according  to  the  rate  of  the  future  time,  by  subduction  of  given  particles  DT  V, 
and  is  therefore  proportional  to  the  time  of  the  whole  ascent.  Q..E.D. 

CASE  2.  If  the  velocity  in  the  ascent 
of  the  body  be  expounded  by  the  length 
AP  as  before,  and  the  resistance  be  made 
as  AP2  -f-  2BAP,and  if  the  force  of  grav 
ity  be  less  than  can  be  expressed  by  DA2 ; 
take  BD  of  such  a  length,  that  AB2  — 
BD2  maybe  proportional  to  the  gravity, 
and  let  DF  be  perpendicular  and  equal 


F  O 


£S2  THE    MATHEMATICAL    PRINCIPLES  [BOOK    ll. 

to  DB,  and  through  the  vertex  F  describe  the  hyperbola  FTVE,  whose  con 
jugate  semi -diameters  are  DB  and  DF;  and  which  cuts  DA  in  E,  and  DP, 
DQ  in  T  and  V ;  and  the  time  of  the  whole  ascent  will  be  as  the  hyper 
bolic  sector  TDE. 

For  the  decrement  PQ  of  the  velocity,  produced  in  a  given  particle  of 
time,  is  as  the  sum  of  the  resistance  AP2  -f  2BAP  and  of  the  gravity 
AB2  —  BD2,  that  is,  as  BP2  —  BD2.  But  the  area  DTV  is  to  the  area 
DPQ  as  DT2  to  DP2 ;  and,  therefore,  if  GT  be  drawn  perpendicular  to 
DF.  as  GT2  or  GD2  —  DF2  to  BD2,  and  as  GD2  to  BP2,  and,  by  di 
vision,  as  DF2  to  BP2  —  BD2.  Therefore  since  the  area  DPQ  is  as  PQ, 
that  is,  as  BP2  —  BD2,  the  area  DTV  will  be  as  the  given  quantity  DF2. 
Therefore  the  area  EDT  decreases  uniformly  in  each  of  the  equal  particles 
of  time,  by  the  subduction  of  so  many  given  particles  DTV,  and  therefore 
is  proportional  to  the  time.  Q.E.D. 

CASE  3.  Let  AP  be  the  velocity  in  the  descent  of 
"""    the  body,  and  AP2  +  2BAP   the  force  of  resistance, 
and  BD2  —  AB2  the  force  of  gravity,  the  angle  DBA 
being  a  right  one.     And  if  with  the  centre  D,  and  the 
principal  vertex  B,  there  be  described   a  rectangular 
hyperbola  BETV  cutting  DA,  DP,  and  DQ  produced 
in  E,  T,  and  V  :  the  sector  DET  of  this  hyperbola  will 
D  be  as  the  whole  time  of  descent. 

For  the  increment  PQ  of  the  velocity,  and  the  area  DPQ  proportional 
to   it,  is  as   the  excess  of   the  gravity  above  the   resistance,  that  is,  as 
m)2?_  AB2  _2BAP  — AP2    or   BD2— BP2.      And  the  area  DTV 
is  to  the  area  DPQ  as  DT3  to  DP2 ;  and  therefore  as  GT2  or  GD"  - 
BD2  to  BP2,  and  as  GD2   to  BD2,  and,  by  division,  as  BD2  to  BD2  - 
BP2.     Therefore  since  the  ami  DPQ  is  as  BD2  —  BP2,  the  area  DTV 
will  be  as  the  given  quantity  BD2.     Therefore  the  area  EDT  increases 
uniformly  in  the  several  equal  particles  of  time  by  the  addition   of  as 
many  given  particles  DTV,  and  therefore  is  proportional   to  the  time  of 
the  descent.     Q.E.D. 

Con.  If  with  the  centre  D  and  the  semi-diameter  DA  there  be  drawn 
through  the  vertex  A  an  arc  A/  similar  to  the  arc  ET,  and  similarly  sub- 
tendino^the  angle  A  DT,  the  velocity  AP  will  be  to  the  velocity  which  the 
body  in  the  time  EDT,  in  a  non-resisting  space,  can  lose  in  its  ascent,  or 
acquire  in  its  descent,  as  the  area  of  the  triangle  DAP  to  the  area  of  the 
Bector  DA/  ;  and  therefore  is  given  from  the  time  given.  For  the  velocity 
ir  a  non-resistin^  medium  is  proportional  to  the  time,  and  therefore  to  this 
sector :  in  a  resisting  medium,  it  is  as  the  triangle ;  and  in  both  mediums, 
where  it  is  least,  it  approaches  to  the  ratio  of  equality,  as  the  sector  and 
triangle  do 


SEC.  III.] 


OF    NATURAL    PHILOSOPHY. 


283 


SCHOLIUM. 

One  may  demonstrate  also  that  case  in  the  ascent  of  the  body,  where  the 
force  of  gravity  is  less  than  can  be  expressed  by  DA2  or  AB2  +  BD2,  and 
greater  than  can  be  expressed  by  AB2  —  DB2,  and  must  be  expressed  by 
AB2.  But  I  hasten  to  other  things 

PROPOSITION  XIV.     THEOREM  XL 

The  same  things  being  supposed,  1  say,  that  the  space  described  in  the 
ascent  or  descent  is  as  the  difference  of  the  area  by  which  the  time  is 
expressed,  and  of  some  other  area  which  is  augmented  or  diminished 
in  an  arithmetical  progression  ;  if  the  forces  compounded  of  the  re 
sistance  and  the  gravity  be  taken,  in  a  geometrical  progression. 
Take  AC  (in  these  three  figures)  proportional  to  the  gravity,  and  AK 

to  the  resistance ;  but  take  them  on  the  same  side  of  the  point  A,  if  the 

\* 

"1 


\ 


B  A 


K  QP 


body  is  descending,  otherwise  on  the  contrary.  Erect  A  b,  which  make  to 
DB  as  DB2  to  4BAC  :  and  to  the  rectangular  asymptotes  CK,  CH,  de 
scribe  the  hyperbola  6N  ;  and,  erecting  KN  perpendicular  to  CK,  the  area 
A/AK  will  be  augmented  or  diminished  in  an  arithmetical  progression, 
while  the  forces  CK  are  taken  in  a  geometrical  progression.  I  say,  there 
fore,  that  the  distance  of  the  body  from  its  greatest  altitude  is  as  the  excess 
of  the  area  A6NK  above  the  area  DET. 

For  since  AK  is  as  the  resistance,  that  is,  as  AP2  X  2BAP  ;  assume 


any  given  quantity  Z,  and  put  AK  equal  to 


then  (by  Lem, 


284 


THE    MATHEMATICAL    PRINCIPLES  [BOOK    II 

2APQ,  +  2B A  X  PU 


II  of  this  Book)  the  moment  KL  of  AK  will  be  equal  to 
2BPQ 


or 


Z 


-,  and  the  moment  KLON  of  the  area  A£NK  will  be  equal  to 


2BPQ  X  LO          BPQ,  X  BD3 
~~Z~~        >r  2Z  X  OK  x~AB" 

CASE  1.  Now  if  the  body  ascends,  and  the  gravity  be  as  AB2  +  BD9 

BET  being    a  circle,  the  line  AC,  which  is  proportional  to  the  gravity 

AW2     i     RF)2 

will  be  -    ~  --  ,  and  DP2  or  AP2  +  2BAP    +  AB2  +  BD2  will  be 

AK  XZ  +  AC  X  Z  or  CK  X  Z  :  and  therefore  the  area  DTV  will  be  to 
the  area  DPQ,  as  DT2  or  I)B2  to  CK  X  Z. 

CASE  2.  If  the  body  ascends,  and  the  gravity  be  as  AB2  —  BD2,  the 
A  r>2  _  HI)2 

line  AC  will  be  "  --  ^  ---  ,  and  DT2  will  be  to  DP2  as  DF2  or  DB2 
Z 

to  BP2—  BD2  or  AP2  +  2BAP  +  AB2  —  BD2,  that  is,  to  AK  X  Z  + 


H 


AC  X  Z  or  CK  X  Z.     And  therefore  the  area  DTV  will  be  to  the  area 
DPQ  as  DB2  to  CK  X  Z. 

CASE  3.  And  by  the  same  reasoning,  if  the  body  descends,  and  therefore 
the   gravity  is    as    BD2  —  AB2,  and   the  line  AC   becomes    equal    to 

or)2 AB2 

5T— r  ;  the  area  DTV  will  be  to  the  area  DPQ  as  DB2  to  CK  X 

Z 

Z  :  as  above. 

Since,  therefore,  these  areas  are  always  in  this  ratio,  if  for  the  area 


SEC.    111^  OF    NATURAL    PHILOSOPHY.  285 

DTY,  by  which  the  moment  of  the  time,  always  equal  to  itself,  is  express 
ed,  there  be  put  any  determinate  rectangle,  as  BD  X  m,  the  area  DPQ,, 
that  is,  |BD  X  PQ,  will  be  to  BD  X  mas  CK  X  Z  to  BD2.  And  thence 
PQ  X  BD3  becomes  equal  to  2  BD  XmX  CK  X  Z,  and  the  moment  KLON 

BP  X  BD  X  tn 
of  the  area  A6NK,  found  before,  becomes  -  .-^  —  --  .     From  the  area 


DET  subduct  its  moment  DTV  or  BD  X  m,  and   there  will  remain 
---  -Pp  —    —  .     Therefore  the  difference  of  the  moments,  that  is,  the 

AP  X  BD  X  m 

mo.nent  of  the  difference  of  tne  areas,  is  equal  to  --  7-5  ---  ;    and 

therefore  (because  of  the  given  quantity  ---  T-~  —  )  as  the  velocity   AP  ; 

that  is,  as  the  moment  of  the  space  which  the  body  describes  in  its  ascent 
or  descent.  And  therefore  the  difference  of  the  areas,  and  that  space,  in 
creasing  or  decreasing  by  proportional  moments,  and  beginning  together  or 
vanishing  together,  are  proportional.  Q.E.D. 

COR.  If  the  length,  which  arises  by  applying  the  area  DET  to  the  line 
BD,  be  called  M  ;  and  another  length  V  be  taken  in  that  ratio  to  the  length 
M,  which  the  line  DA  has  to  the  line  DE;  the  space  which  a  body,  in  a 
resisting  medium,  describes  in  its  whole  ascent  or  descent,  will  be  to  the 
space  which  a  body,  in  a  non-resisting  medium,  falling  from  rest,  can  de 
scribe  in  the  same  time,  as  the  difference  of  the  aforesaid  areas  to 

BD  X  V2 
-  -TO""""  j  an(^  therefore  is  given  from  the  time  given.     For  the  space  in  a 

A.LJ 

non-resisting  medium  is  in  a  duplicate  ratio  of  the  time,  or  as  V2  ;  and. 

BD  X  V2 
because  BD  and  AB  are  given,  as  ---  -TTT-  —  .     This  area  is  equal  to  the 

DA2  X  BD  x  M2 
area  --  fvGr*~~~T~R  —  "~  anc*  ^ne  moment  Of   M  is  m  ;  and  therefore  the 

DA2  X  BD  X  2M  X  m 

moment  of  this  area  is  ---  =-  ---  ^5  —    -.     But  this  moment  is  to 

""  X  .A  D 


the  moment  of  the  difference  of  the  aforesaid  areas  DET  and  A6NK,  viz.,  to 

AP  X  BD  X  m      DA2  X  BD  X  Mx  DA2  .       ^^ 

-      --  ,  as  -      -r-     -  to  |BD  X  AP,  or  as          into  DET 


to  DAP  ;  and,  therefore,  when  the  areas  DKT  and  DAP  are  least,  in  the 

BD  X  V2 
ratio  of  equality.     Therefore  the  area  —  r^  --  and  the  difference  of  the 

areas  DET  and  A&NK,  when  all  these  areas  are  least,  have  equal  moments  ; 
and  {  re  therefore  equal.  Therefore  since  the  velocities,  and  therefore  also 
the  s]  aces  in  both  mediums  described  together,  in  the  beginning  of  the  de 
scent  or  the  end  of  the  ascent,  approach  to  equality,  and  therefore  are  then 


286  THE    MATHEMATICAL    PRINCIPLES  [BOOK    II 

BD  X  V2 

one  to  another  as  the  area r-^ — ,  and  the  difference  of  the  areas  DET 

AD 

and  A6NK ;  and  moreover  since  the  space,  in  a  non-resisting  medium,  is 

BD  X  V2 

perpetually  as Tu~~>  an(^  tne  sPace>  in  a  resisting  medium,  is  perpetu 
ally  as  the  difference  of  the  areas  DET  and  A&NK  ;  it  necessarily  follows, 
that  the  spaces,  in  both  mediums,  described  in  any  equal  times,  are  one  to 

BD  X  V2 
another  as  that  area 7-5 — »  an(^  ^he  difference  of  the  areas  DET  and 

A6NK.     Q.E.D. 

SCHOLIUM. 

The  resistance  of  spherical  bodies  in  fluids  arises  partly  from  the  tena 
city,  partly  from  the  attrition,  and  partly  from  the  density  of  the  medium. 
And  that  part  of  the  resistance  which  arises  from  the  density  of  the  fluid 
is,  as  I  said,  in  a  duplicate  ratio  of  the  velocity ;  the  other  part,  which 
arises  from  the  tenacity  of  the  fluid,  is  uniform,  or  as  the  moment  of  the 
time  ;  and,  therefore,  we  might  now  proceed  to  the  motion  of  bodies,  whicli 
are  resisted  partly  by  an  uniform  force,  or  in  the  ratio  of  the  moments  of 
the  time,  and  partly  in  the  duplicate  ratio  of  the  velocity.  But  it  is  suf 
ficient  to  have  cleared  the  way  to  this  speculation  in  Prop.  VIII  and  IX 
foregoing,  and  their  Corollaries.  For  in  those  Propositions,  instead  of  the 
uniform  resistance  made  to  an  ascending  body  arising  from  its  gravity, 
one  may  substitute  the  uniform  resistance  which  arises  from  the  tenacity 
of  the  medium,  when  the  body  moves  by  its  vis  insita  alone  ;  and  when  the 
body  ascends  in  a  right  line,  add  this  uniform  resistance  to  the  force  of 
gravity,  and  subduct  it  when  the  body  descends  in  a  right  line.  One 
might  also  go  on  to  the  motion  of  bodies  which  are  resisted  in  part  uni 
formly,  in  part  in  the  ratio  of  the  velocity,  and  in  part  in  the  duplicate 
ratio  of  the  same  velocity.  And  I  have  opened  a  way  to  this  in  Prop. 
XIII  and  XIV  foregoing,  in  which  the  uniform  resistance  arising  from  the 
tenacity  of  the  medium  may  be  substituted  for  the  force  of  gravity,  or  be 
compounded  with  it  as  before.  But  I  hasten  to  other  things. 


SKC.    -IV'.]  OF    NATUEAL    PHILOSOPHY.  2S? 

SECTION  IV. 

Of  the  circular  motion  of  bodies  in  resisting'  mediums. 

LEMMA  III. 

Let  PQR  be  a  spiral  rutting  all  the  radii  SP,  SO,  SR,  <J*c.,  in  equal 
angles.     Draw  tfie  right  line  PT  touching  the  spiral  in  any  point  P, 
and  cutting   the  radius   SQ  in  T  ;  cfo'er?0  PO,  QO  perpendicular  to 
the  spiral,  and  meeting-  in,  O,  and  join  SO.     .1  say,  that  if  Hie  points 
P  a/*(/  Q  approach  and  coincide,  the  angle  PSO  vri/Z  become  a  right 
angle,  and  the  ultimate  ratio  of  the  rectangle  TQ,  X  2PS  to  P^3  //>i// 
/>e  £/ie  ya/io  o/"  equality. 
For  from  the  right  angles  OPQ,  OQR,  sub 
duct  the  equal  angles  SPQ,  SQR,  and  there 
will   remain  the  equal  angles    OPS,   OQS. 
Therefore  a  circle  which  passes  through  the 
points  OSP  will  pass  also  through  the  point 
Q.     Let  the  points  P  and  Q,  coincide,  and 
this  circle  will  touch  the  spiral  in  the  place 
of  coincidence  PQ,  and  will  therefore  cut  the 
right  line  OP  perpendicularly.     Therefore  OP  will  become  a  diameter  of 
this  circle,  and  the  angle  OSP,  being  in  a  semi-circle,  becomes   a  right 
one.     Q.E.1). 

Draw  Q,D,  SE  perpendicular  to  OP,  and  the  ultimate  ratios  of  the  lines 
will  be  as  follows  :  TO  to  PD  as  TS  or  PS  to  PE,  or  2PO  to  2PS  •  and 
PD  to  PO  as  PO  to  2PO ;  and,  ex  cequo  pertorbatt,  to  TO  to  PO  as  PO 
to  2PS.  Whence  PO2  becomes  equal  to  TO  X  2PS.  O.E.D. 

PROPOSITION  XV.     THEOREM  XII. 

Tf  the  density  of  a  medium  in  each  place  thereof  be  recipr on iJl y  as  the 
distance  of  the  places  from  an  immovable  centre,  aud  the  centripetal 
force  be  in  the  duplicate  ratio  of  the  density  ;  I  say,  that  a  body  mny 
revolve  in  a  spiral  which  cuts  all  the  radii  drawn  from  that  centre 
in  a  given  angle. 

Suppose  every  thing  to  be  as  in  the  forego 
ing  Lemma,  and  produce  SO  to  V  so  that  SV 
may  be  equal  to  SP.  In  any  time  let  a  body, 
in  a  resisting  medium,  describe  the  least  arc 
PO,  and  in  double  the  time  the  least  arc  PR : 
and  the  decrements  of  those  arcs  arising  from 
the  resistance,  or  their  differences  from  the 
arcs  which  would  be  described  in  a  non-resist 
ing  medium  in  the  same  times,  will  be  to  each 
other  as  the  squares  of  the  times  in  which  they 
are  generated  ;  therefore  the  decrement  of  the 


288  THE    MATHEMATICAL    PRINCIPLES  [_BoOK    11 

arc  PQ  is  the  fourth  part  of  the  decrement  of  the  arc  PR.  Whence  also 
if  the  area  QSr  be  taken  equal  to  the  area  PSQ,  the  decrement  of  the  arc 
PQ  will  be  equal  to  half  the  lineola  Rr  ;  and  therefore  the  force  of  resist 
ance  and  the  centripetal  force  are  to  each  other  as  the  lineola  jRrandTQ 
which  they  generate  in  the  same  time.  Because  the  centripetal  force  with 
which  the  body  is  urged  in  P  is  reciprocally  as  SP2,  and  (by  Lem.  X, 
Book  I)  the  lineola  TQ,  which  is  generated  by  that  force,  is  in  a  ratio 
compounded  of  the  ratio  of  this  force  and  the  duplicate  ratio  of  the  time 
in  which  the  arc  PQ,  is  described  (for  in  this  case  I  neglect  the  resistance, 
as  being  infinitely  less  than  the  centripetal  force),  it  follows  that  TQ  X 
SP2,  that  is  (by  the  last  Lemma),  fPQ2  X  SP,  will  be  in  a  duplicate  ra 
tio  of  the  time,  and  therefore  the  time  is  as  PQ,  X  v/SP  ;  and  the  velo 
city  of  the  body,  with  which  the  arc  PQ  is  described  in  that  time,  as 

PQ  1 

—  -«p  or  ,  that  is,  in  the  subduplicate  ratio  of  SP  reciprocally. 


And,  by  a  like  reasoning,  the  velocity  with  which  the  arc  QR  is  described, 
is  in  the  subduplicate  ratio  of  SQ  reciprocally.  Now  those  arcs  PQ  and 
QR  are  as  the  describing  velocities  to  each  other  ;  that  is,  in  the  subdu 


plicate  ratio  of  SQ  to  SP,  or  as  SQ  to  x/SP  X  SQ;  and,  because  of  the 
equal  angles  SPQ,  SQ?',  and  the  equal  areas  PSQ,  QSr,  the  arc  PQ  is  to 
the  arc  Qr  as  SQ  to  SP.  Take  the  differences  of  the  proportional  conse 
quents,  and  the  arc  PQ  will  be  to  the  arc  Rr  as  SQ  to  SP  —  VSP  X  ~SQ~, 
or  ^VQ.  For  the  points  P  and  Q  coinciding,  the  ultimate  ratio  of  SP  — 


X  SQ  to  |VQ  is  the  ratio  of  equality.     Because  the  decrement  of 
the  arc  PQ  arising  from  the  resistance,  or  its  double  Rr,  is  as  the  resistance 

and  the  square  of  the  time  conjunctly,  the  resistance  will  be  &Sp-^r  —  op. 

*1 


X 
But  PQ  was  to  Rr  as  SQ  to  |VQ,  and  thence  SSaTXToD  becomes  as 

Jr  vst      X  oJr 

-VQ  -OS 

pWxsvxSQ,'  or  ns  ETp^TsP-     For  the  poillts  p  and  a  coincidin& 

SP  and  SQ  coincide  also,  and  the  angle  PVQ  becomes  a  right  one;  and, 
because  of  the  similar  triangles  PVQ,  PSO,  PQ.  becomes  to   '-VQ  as  OP 

OS 

to  |  OS.     Therefore    :  y  —  —  --  is  as  the  resistance,  that  is,  in  the  ratio  of 
\J  i    X  ol 

the  density  of  the  medium  in  P  and  the  duplicate  ratio  of  the  velocity 
conjunc-tly.     Subduct  the  duplicate  ratio  of  the  velocity,  namely,  the  ratio 

1  OS 

^5,  and  there  will  remain  the  density  of  the  medium  in  P.  as  7^5  -  = 

OA  Ur    X    fei 

Let  the  spiral  be  given,  and;  because  of  the  given  ratio  of  OS  to  OP,  the 
density  of  the  medium  in  P  will  be  as  ~-p.     Therefore  in  a  medium  whose 


SEC.  IV,]  OF    NATURAL    PHILOSOPHY.  2S9 

density  is  reciprocally  as  SP  the  distance  from  the  centre,  a  body  will  re 
volve  in  this  spiral.  Q.E.D. 

COR.  1.  The  velocity  in  any  place  P,  is  always  the  same  wherewith  a 
body  in  a  non-resisting  medium  with  the  same  centripetal  force  would  re 
volve  in  a  circle,  at  the  same  distance  SP  from  the  centre. 

COR.  2.  The  density  of  the  medium,  if  the  distance  SP  be  given,  is  as 

OS  OS 

TTp,  but  if  that  distance  is  not  given,  as  ^ ^5.     And   thence  a  spiral 

may  be  fitted  to  any  density  of  the  medium. 

COR.  3.  The  force  of  the  resistance  in  any  place  P  is  to  the  centripetal 
force  in  the  same  place  as  |OS  to  OP.  For  those  forces  are  to  each  other 

^VQ  x  PQ         iPQ2 

as  iRr  and  TQ,  or  as  1—^-^~-  and  ^-,  that  is,  as  iVQ  and  PQ, 

ol%,  ol 

or  |OS  and  OP.  The  spiral  therefore  being  given,  there  is  given  the  pro 
portion  of  the  resistance  to  the  centripetal  force  ;  and,  vice  versa,  from  that 
proportion  given  the  spiral  is  given. 

COR.  4.  Therefore  the  body  cannot  revolve  in  this  spiral,  except  where 
the  force  of  resistance  is  less  than  half  the  centripetal  force.  Let  the  re 
sistance  be  made  equal  to  half  the  centripetal  force,  and  the  spiral  will  co 
incide  with  the  right  line  PS,  and  in  that  right  line  the  body  will  descend 
to  the  centre  with  a  velocity  that  is  to  the  velocity,  with  which  it  was 
proved  before,  in  the  case  of  the  parabola  (Theor.  X,  Book  I),  the  descent 
would  be  made  in  a  non-resisting  medium,  in  the  subduplicate  ratio  of 
unity  to  the  number  two.  And  the  times  of  the  descent  will  be  here  recip 
rocally  as  the  velocities,  and  therefore  given. 

COR.  5.  And  because  at  equal  distances 
from  the  centre  the  velocity  is  the  same  in  the 
spiral  PQ,R  as  it  is  in  the  right  line  SP,  and 
the  length  of  the  spiral  is  to  the  length  of  the 
right  line  PS  in  a  given  ratio,  namely,  in  the 
ratio  of  OP  to  OS  ;  the  time  of  the  descent  in 
the  spiral  will  be  to  the  time  of  the  descent  in 
the  right  line  SP  in  the  same  given  ratio,  and 
therefore  given. 

COR.  6.  If  from  the  centre  S,  with  any  two 
given  intervals,  two  circles  are  described ;  and 
these  circles  remaining,  the  angle  which  the  spiral  makes  with  the  radius" 
PS  be  any  how  changed ;  the  number  of  revolutions  which  the  body  can 
complete  in  the  space  between  the  circumferences  of  those  circles,  going 

PS 

round  in  the  spiral  from  one  circumference  to  another,  will  be  as  7^,  or  as 

Ok5' 

ths  tangent  of  the  angle  which  the  spiral  makes  with  the  radius  PS  ;  and 

19 


290  THE    MATHEMATICAL    PRINCIPLES  [BOOK    II 

OP 

the  time  of  the  same  revolutions  will  be  as  -^,  that  is,  as  the  secant  of  the 

Uo 

same  angle,  or  reciprocally  as  the  density  of  the  medium. 

COR.  7.  If  a  body,  in  a  medium  whose  density  is  reciprocally  as  the  dis 
tances  of  places  from  the  centre,  revolves  in  any  curve  AEB  about  that 
centre,  and  cuts  the  first  radius  AS  in  the  same 
angle  in  B  as  it  did  before  in  A,  and  that  with  a 
velocity  that  shall  be  to  its  first  velocity  in  A  re 
ciprocally  in  a  subduplicate  ratio  of  the  distances 
from  the  centre  (that  is,  as  AS  to  a  mean  propor 
tional  between  AS  and  BS)  that  body  will  con 
tinue  to  describe  innumerable  similar  revolution? 
BFC,  CGD,  &c.,  and  by  its  intersections  will 
distinguish  the  radius  AS  into  parts  AS,  BS,  CS,  DS,  &c.,  that  are  con 
tinually  proportional.  But  the  times  of  the  revolutions  will  be  as  the 
perimeters  of  the  orbits  AEB,  BFC,  CGD,  &c.,  directly,  and  the  velocities 

3  3 

at  the  beginnings  A,  B,  C  of  those  orbits  inversely  ;  that  is  as  AS  %   BS  % 

CS"2".     And  the  whole  time  in  which  the  body  will  arrive  at  the  centre, 
will  be  to  the  time  of  the  first  revolution  as  the  sum  of  all  the  continued 

142 

proportionals  AS2,   BS2,   CS2,  going  on  ad  itifinitum,  to  the  first  term 

*  i  3 

AS2  ;  that  is,  as  the  first  term  AS2  to  the  difference  of  the  two  first  AS2 

—  BS%  or  as  f  AS  to  AB  very  nearly.     Whence  the  whole  time  may  be 
easily  found. 

COR.  8.  From  hence  also  may  be  deduced,  near  enough,  the  motions  of 
bodies  in  mediums  whose  density  is  either  uniform,  or  observes  any  other 
assigned  law.  From  the  centre  S,  with  intervals  SA,  SB,  SC,  &c.,  con 
tinually  proportional,  describe  as  many  circles  ;  and  suppose  the  time  of 
the  revolutions  between  the  perimeters  of  any  two  of  those  circles,  in  the 
medium  whereof  we  treated,  to  be  to  the  time  of  the  revolutions  between 
the  same  in  the  medium  proposed  as  the  mean  density  of  the  proposed  me 
dium  between  those  circles  to  the  mean  density  of  the  medium  whereof  we 
treated,  between  the  same  circles,  nearly  :  and  that  the  secant  of  the  angle 
in  which  the  spiral  above  determined,  in  the  medium  whereof  we  treated, 
cuts  the  radius  AS,  is  in  the  same  ratio  to  the  secant  of  the  angle  in  which 
the  new  spiral,  in  the  proposed  medium,  cuts  the  same  radius :  and  also 
that  the  number  of  all  the  revolutions  between  the  same  two  circles  is  nearly 
as  the  tangents  of  those  angles.  If  this  be  done  every  where  between  every 
two  circles,  the  motion  will  be  continued  through  all  the  circles.  And  by 
this  means  one  may  without  difficulty  conceive  at  what  rate  and  in  what 
time  bodies  ought  to  revolve  in  any  regular  medium. 


•SEC.    IY.1  OF    NATURAL    PHILOSOPHY.  291 

COR.  9.  And  although  these  motions  becoming  eccentrical  should  be 
performed  in  spirals  approaching  to  an  oval  figure,  yet,  conceiving  the 
several  revolutions  of  those  spirals  to  be  at  the  same  distances  from  each 
other,  and  to  approach  to  the  centre  by  the  same  degrees  as  the  spiral  above 
described,  we  may  also  understand  how  the  motions  of  bodies  may  be  per 
formed  in  spirals  of  that  kind. 

PROPOSITION  XVI.     THEOREM  XIII. 

If  the  density  of  the  medium  in  each  of  the  places  be  reciprocally  as  the 
distance  of  the  >,  places  from  the  immoveable  centre,  and  the  centripetal 
force  be  reciprocally  as  any  power  of  the  same  distance,  I  say,  that  the 
body  may  revolve  in  a  spiral  intersecting  all  the  radii  drawn  from 
that  centre  in  a  given,  angle. 
This  is  demonstrated  in  the  same  manner  as 

the  foregoing  Proposition.     For  if  the  centri 

petal  force  in  P  be  reciprocally  as  any  power 

SPn+1  of  the  distance  SP  whose  index  is  n 

+  1  ;  it  will  be  collected,  as  above,  that  the 

time  in  which  the  body  describes  any  arc  PQ, 

i 
will  be  as  PQ,  X  PS2U  ;  and  the  resistance  in 


i!!  x      _ 

n;°raS  "~' 


X  SPn;PQ,  X  SP"XSQ,' 


,       ,  1  —  in  X   OS  .  1  —  \n  X  OS     . 

therefore  as  Qp"^~gpirqTT'  tliat  1S>  (because  -    ~~Qp~~          1S    a    £lven 

quantity),  reciprocally  as  SPn+  ! .    And  therefore,  since  the  velocity  is  recip 
rocally  as  SP3",  the  density  in  P  will  be  reciprocally  as  SP. 

COR.  1.  The  resistance  is  to  the  centripetal  force  as  1  —  ^//.  X  OS 
to  OP. 

COR.  2.  If  the  centripetal  force  be  reciprocally  as  SP3.  1  —  ±w  will  be 
===  0 ;  and  therefore  the  resistance  and  density  of  the  medium  will  be 
nothing,  as  in  Prop.  IX,  Book  I. 

COR.  3.  If  the  centripetal  force  be  reciprocally  as  any  power  of  the  ra 
dius  SP,  whose  index  is  greater  than  the  number  3,  the  affirmative  resist 
ance  will  be  changed  into  a  negative. 

SCHOLIUM. 

This  Proposition  and  the  former,  which  relate  to  mediums  of  unequal 
density,  are  to  be  understood  of  the  motion  of  bodies  that  are  so  small,  that 
the  greater  density  of  the  medium  on  one  side  of  the  body  above  that  on 
the  other  is  not  to  be  considered.  I  suppose  also  the  resistance,  cateris 
paribus,  to  be  proportional  to  its  density.  Whence,  in  mediums  whose 


292  THE    MATHEMATICAL    PRINCIPLES  |  BoOK    II 

force  of  resistance  is  not  as  the  density,  the  density  must  be  so  much  aug 
mented  or  diminished,  that  either  the  excess  of  the  resistance  may  be  taken 
away,  or  the  defect  supplied. 

PROPOSITION  XVII.    PROBLEM  IV 

To  find  the  centripetal  force  and  the  resisting  force  of  the  medium,  by 
which  a  body,  the  law  of  the  velocity  being  given,  shall  revolve  in  a 
given  spiral. 

Let  that  spiral  be  PQR.  From  the  velocity, 
with  which  the  body  goes  over  the  very  small  arc 
PQ,,  the  time  will  be  given :  and  from  the  altitude 
TQ,,  which  is  as  the  centripetal  force,  and  the 
square  of  the  time,  that  force  will  be  given.  Then 
from  the  difference  RSr  of  the  areas  PSQ,  and 
Q,SR  described  in  equal  particles  of  time,  the  re 
tardation  of  the  body  will  be  given ;  and  from 
the  retardation  will  be  found  the  resisting  force 
and  density  of  the  medium. 

PROPOSITION  XVIII.     PROBLEM  V. 

The  law  of  centripptal  force  being  given,  to  find  the  density  of  the  me 
dium  in  each  of  the  places  thereof,  by  which  a  body  may  describe  a 
given  spiral. 

From  the  centripetal  force  the  velocity  in  each  place  must  be  found  ; 
then  from  the  retardation  of  the  velocity  the  density  of  the  medium  is 
found,  as  in  the  foregoing  Proposition. 

But  I  have  explained  the  method  of  managing  these  Problems  in  the 
tenth  Proposition  and  second  Lemma  of  this  Book;  and  will  no  longer 
detain  the  reader  in  these  perplexed  disquisitions.  I  shall  now  add  some 
things  relating  to  the  forces  of  progressive  bodies,  and  to  the  density  and 
resistance  of  those  mediums  in  which  the  motions  hitherto  treated  of,  and 
those  akin  to  them,  are  performed. 


SEC.    V.]  OF    NATURAL    PHILOSOPHY.  293 

SECTION  V. 

Of  the  density  and  compression  of  fluids  ;  and  of  hydrostatics. 

THE    DEFINITION    OF    A    FLUID. 

A  fluid  is  any  body  whose  parts  yield  to  any  force  impressed  on  it, 
by  yielding,  are  easily  moved  among  themselves. 

PROPOSITION  XIX.     THEOREM  XIv 

All  the  parts  of  a  homogeneous  and  unmoved  fluid  included  in  any  un 
moved  vessel,  and  compressed  on  every  side  (setting  aside  the  consider 
ation  of  condensation,   gravity,   and   all  centripetal  forces],   will  be 
equally  pressed  on  every  side,  and  remain  in  their  places  without  any 
motion  arising  from  that  pressure. 
CASE  1.  Let  a  fluid  be  included  in  the  spherical 
vessel   ABC,  and  uniformly  compressed  on  every 
side :  1  say,  that  no  part  of  it  will  be  moved  by 
that  pressure.     For  if  any  part,  as  L),  be  moved, 
all  such  parts  at  the  same  distance  from  the  centre 
on   every  side  must  necessarily  be  moved   at   the 
same  time  by  a  like  motion ;  because  the  pressure 
of  them  all  is  similar  and  equal ;  and  all  other 
motion  is  excluded  that  does  not  arise  from  that 
pressure.     But  if  these  parts  come  all  of  them  nearer  to  the  centre,  the 
fluid   must  be  condensed  towards  the  centre,  contrary  to  the  supposition. 
If  they  recede  from  it,  the  fluid  must  be  condensed  towards  the  circumfer 
ence  ;  which  is  also  contrary  to  the  supposition.     Neither   can  they  move 
in  any  one  direction  retaining  their  distance   from  the  centre,  because  for 
the  same  reason,  they  may  move  in  a  contrary  direction  :  but   the  sami 
part  cannot  be  moved   contrary  ways  at  the  same   time.     Therefore  no 
part  of  the  fluid  will  be  moved  from  its  place.     Q,.E.T). 

CASE  2.  I  say  now,  that  all  the  spherical  parts  of  this  fluid  are  equally 
pressed  on  every  side.  For  let  EF  be  a  spherical  part  of  the  fluid  ;  if  this 
be  not  pressed  equally  on  every  side,  augment  the  lesser  pressure  till  it  be 
pressed  equally  on  every  side;  and  its  parts  (by  Case  I)  will  remain  in 
their  places.  But  before  the  increase  of  the  pressure,  they  would  remain 
in  their  places  (by  Case  1) ;  and  by  the  addition  of  a  new  pressure  they 
will  be  moved,  by  the  definition  of  a  fluid,  from  those  places.  Now  these 
two  conclusions  contradict  each  other.  Therefore  it  was  false  to  say  that 
the  sphere  EF  was  not  pressed  equally  on  every  side.  Q,.E.D. 

CASE  3.  I  say  besides,  that  different  spherical  parts  have  equal  pressures. 
For  the  contiguous  spherical  parts  press  each  other  mutually  and  equally 
in  the  point  of  contact  (by  Law  III).  But  (by  Case  2)  they  are  pressed  on 
every  side  with  the  same  force.  Therefore  any  two  spherical  parts  lot 


391  THE    MATHEMATICAL    PRINCIPLES  [BoOK    II. 

contiguous,  since  an  intermediate  spherical  part  can  touch  both,  will  be 
pressed  with  the  same  force.  Q.E.D. 

CASE  4.  I  say  now,  that  all  the  parts  of  the  fluid  are  every  where  press 
ed  equally.  For  any  two  parts  may  be  touched  by  spherical  parts  in  any 
points  whatever ;  and  there  they  will  equally  .press  those  spherical  parts 
(by  Case  3).  and  are  reciprocally  equally  pressed  by  them  (by  Law  III). 
Q.E.D. 

CASE  5.  Since,  therefore,  any  part  GHI  of  the  fluid  is  inclosed  by  the 
rest  of  the  fluid  as  in  a  vessel,  and  is  equally  pressed  on  every  side  ;  and 
also  its  parts  equally  press  one  another,  and  are  at  rest  among  themselves ; 
it  is  manifest  that  all  the  parts  of  any  fluid  as  GHI,  which  is  pressed 
equally  on  every  side,  do  press  each  other  mutually  and  equally,  and  are  at 
rest  among  themselves.  Q.E.D. 

CASE  6.  Therefore  if  that  fluid  be  included  in  a  vessel  of  a  yielding 
substance,  or  that  is  not  rigid,  and  be  not  equally  pressed  on  every  side, 
the  same  will  give  way  to  a  stronger  pressure,  by  the  Definition  of  fluidity. 

CASE  7.  And  therefore,  in  an  inflexible  or  rigid  vessel,  a  fluid  will  not 
Sustain  a  stronger  pressure  on  one  side  than  on  the  other,  but  will  give 
way  to  it,  and  that  in  a  moment  of  time ;  because  the  rigid  side  of  the 
vessel  does  not  follow  the  yielding  liquor.  But  the  fluid,  by  thus  yielding, 
will  press  against  the  opposite  side,  and  so  the  pressure  will  tend  on  every 
side  to  equality.  And  because  the  fluid,  as  soon  as  it  endeavours  to  recede 
from  the  part  that  is  most  pressed,  is  withstood  by  the  resistance  of  the 
vessel  on  the  opposite  side,  the  pressure  will  on  every  side  be  reduced  to 
equality,  in  a  moment  of  time,  without  any  local  motion  :  and  from  thence 
the  parts  of  the  fluid  (by  Case  5)  will  press  each  other  mutually  and  equal 
ly,  and  be  at  rest  among  themselves.  Q..E.D. 

COR.  Whence  neither  will  a  motion  of  the  parts  of  the  fluid  among 
themselves  be  changed  by  a  pressure  communicated  to  the  external  super 
ficies,  except  so  far  as  either  the  figure  of  the  superficies  may  be  somewhere 
altered,  or  that  all  the  parts  of  the  fluid,  by  pressing  one  another  more  in 
tensely  or  remissly,  may  slide  with  more  or  less  difficulty  among  them 
selves. 

PROPOSITION  XX.     THEOREM  XV. 

Jf  all  the  parts  of  a  spherical  fluid,  homogeneous  at  equal  distances  from 
the  centre,  lying  on.  a  spherical  concentric  bottom,  gravitate  towards 
the  centre  of  the  whole,  the  bottom  will  sustain  the  weight  of  a  cylin 
der,  whose  base  is  equal  to  the  superficies  of  the  bottom,  and  whose  al 
titude  is  the  same  with  that  of  the  incumbent  fluid. 
Let  OHM  be  the  superficies  of  the  bottom,  and  AEI  the  upper  super 
ficies  of  the  fluid.     Let  the  fluid  be  distinguished  into  concentric  orbs  of 
3qual  thickness,  by  the  innumerable  spherical  superficies  *3PK,  CGL  :  and 


SEC.    V  OF    NATURAL    PHILOSOPHY.  295 

conceive  the  force  of  gravity  to  act  only  in  the 
upper  superficies  of  every  orb,  and  the  actions 
to  be  equal  on  the  equal  parts  of  all  the  su 
perficies.  Therefore  the  upper  superficies  AE 
is  pressed  by  the  single  force  of  its  own  grav 
ity,  by  which  all  the  parts  of  the  upper  orb, 
and  the  second  superficies  BFK,  will  (by 
Prop.  XIX),  according  to  its  measure,  be 
equally  pressed.  The  second  superficies  BFK 
is  pressed  likewise  by  the  force  of  its  own 
gravity,  which,  added  to  the  former  force, 
makes  the  pressure  double.  The  third  superficies  CGL  is,  according  to  its 
measure,  acted  on  by  this  pressure  and  the  force  of  its  own  gravity  besides, 
which  makes  its  pressure  triple.  And  in  like  manner  the  fourth  superfi 
cies  receives  a  quadruple  pressure,  the  fifth  superficies  a  quintuple,  and  so 
on.  Therefore  the  pressure  acting  on  every  superficies  is  not  as  the  solid 
quantity  of  the  incumbent  fluid,  but  as  the  number  of  the  orbs  reaching 
to  the  upper  surface  of  the  fluid  ;  and  is  equal  to  the  gravity  of  the  lowest 
orb  multiplied  by  the  number  of  orbs :  that  is,  to  the  gravity  of  a  solid 
whose  ultimate  ratio  to  the  cylinder  above-mentioned  (when  the  number  of 
the  orbs  is  increased  and  their  thickness  diminished,  ad  infiititum,  so  that 
the  action  of  gravity  from  the  lowest  superficies  to  the  uppermost  may  be- 
some  continued)  is  the  ratio  of  equality.  Therefore  the  lowest  superficies 
•sustains  the  weight  of  the  cylinder  above  determined.  Q..E.D.  And  by  a 
like  reasoning  the  Proposition  will  be  evident,  where  the  gravity  of  the 
fluid  decreases  in  any  assigned  ratio  of  the  distance  from  the  centre,  and 
also  where  the  fluid  is  more  rare  above  and  denser  below.  Q.E.D. 

COR.  1.  Therefore  the  bottom  is  not  pressed  by  the  whole  weight  of  the 
incumbent  fluid,  but  only  sustains  that  part  of  it  which  is  described  in  the 
Proposition  ;  the  rest  of  the  weight  being  sustained  archwise  by  the  spheri 
cal  figure  of  the  fluid. 

COR.  2.  The  quantity  of  the  pressure  is  the  same  always  at  equal  dis 
tances  from  the  centre,  whether  the  superficies  pressed  be  parallel  to  the 
horizon,  or  perpendicular,  or  oblique ;  or  whether  the  fluid,  continued  up 
wards  from  the  compressed  superficies,  rises  perpendicularly  in  a  rectilinear 
direction,  or  creeps  obliquely  through  crooked  cavities  and  canals,  whether 
those  passages  be  regular  or  irregular,  wide  or  narrow.  That  the  pressure 
is  not  altered  by  any  of  these  circumstances,  may  be  collected  by  applying 
the  demonstration  of  this  Theorem  to  the  several  cases  of  fluids. 

COR.  3.  From  the  same  demonstration  it  may  also  be  collected  (by  Prop. 
XIX),  that  the  parts  of  a  heavy  fluid  acquire  no  motion  among  themselvei 
by  the  pressure  of  the  incumbent  veight,  except  that  motion  which  arises 
from  condensation. 


296  THE    MATHEMATICAL    PRINCIPLES  [BCOK    II 

Con.  4.  And  therefore  if  another  body  of  the  same  specific  gravity,  in 
capable  of  condensation,  be  immersed  in  this  fluid,  it  will  acquire  no  mo 
tion  by  the  pressure  of  the  incumbent  weight:  it  will  neither  descend  nor  . 
ascend,  nor  change  its  figure.  If  it  be  spherical,  it  will  remain  so,  notwith 
standing  the  pressure  ;  if  it  be  square,  it  will  remain  square;  and  that, 
whether  it  be  soft  or  fluid :  whether  it  swims  freely  in  the  fluid,  or  lies  at 
the  bottom.  For  any  internal  part  of  a  fluid  is  in  the  same  state  with  the 
submersed  body  ;  and  the  case  of  all  submersed  bodies  that  have  the  same 
magnitude,  figure,  and  specific  gravity,  is  alike.  If  a  submersed  body,  re 
taining  its  weight,  should  dissolve  and  put  on  the  form  of  a  fluid,  this 
body,  if  before  it  would  have  ascended,  descended,  or  from  any  pressure  as 
sume  a  new  figure,  would  now  likewise  ascend,  descend,  or  put  on  a  new 
figure  ;  and  that,  because  its  gravity  and  the  other  causes  of  its  motion 
remain.  But  (by  Case  5,  Prop.  XIX;  it  would  now  be  at  rest,  and  retain 
its  figure.  Therefore  also  in  the  former  case. 

COR.  5.  Therefore  a  body  that  is  specifically  heavier  than  a  fluid  con 
tiguous  to  it  will  sink ;  and  that  which  is  specifically  lighter  will  ascend, 
and  attain  so  much  motion  and  change  of  figure  as  that  excess  or  defect  of 
gravity  is  able  to  produce.  For  that  excess  or  defect  is  the  same  thing  as  an 
impulse,  by  which  a  body,  otherwise  in  equilibria  with  the  parts  of  the 
fluid,  is  acted  on:  and  may  be  compared  with  the  excess  or  defect  of  a 
weight  in  one  of  the  scales  of  a  balance. 

COR.  6.  Therefore  bodies  placed  in  fluids  have  a  twofold  gravity  •  the 
one  true  and  absolute,  the  other  apparent,  vulgar,  and  comparative.  Ab 
solute  gravity  is  the  whole  force  with  which  the  body  tends  downwards ; 
relative  and  vulgar  gravity  is  the  excess  of  gravity  with  which  the  body 
tends  downwards  more  than  the  ambient  fluid.  By  the  first  kind  of  grav 
ity  the  parts  of  all  fluids  and  bodies  gravitate  in  their  proper  places ;  and 
therefore  their  weights  taken  together  compose  the  weight  of  the  whole. 
For  the  whole  taken  together  is  heavy,  as  may  be  experienced  in  vessels 
full  of  liquor  ;  and  the  weight  of  the  whole  is  equal  to  the  weights  of  all 
the  parts,  and  is  therefore  composed  of  them.  By  the  other  kind  of  grav 
ity  bodies  do  not  gravitate  in  their  places ;  that  is,  compared  with  one 
another,  they  do  not  preponderate,  but,  hindering  one  another's  endeavours 
to  descend,  remain  in  their  proper  places,  as  if  they  were  not  heavy.  Those 
things  which  are  in  the  air,  and  do  not  preponderate,  are  commonly  looked 
on  as  not  heavy.  Those  which  do  preponderate  are  commonly  reckoned 
heavy,  in  as  much  as  they  are  not  sustained  by  the  weight  of  the  air.  The 
Common  weights  are  nothing  else  but  the  excess  of  the  true  weights  above 
the  weight  of  the  air.  Hence  also,  vulgarly,  those  things  are  called  light 
which  are  less  heavy,  and,  by  yielding  to  the  preponderating  air,  mount 
upwards.  But  these  are  only  comparatively  lig  s  &mA  not  truly  so,  because 

hey  descend  in  racuo.     Thus,  in  water,  bodies  *>icfc.  by  their  greater  or 


SEC.    V.]  OF    NATURAL    PHILOSOPHY.  297 

less  gravity,  descend  or  ascend,  are  comparatively  and  apparently  heavy  or 
light ;  and  their  comparative  and  apparent  gravity  or  levity  is  the  excess 
.or  defect  by  which  their  true  gravity  either  exceeds  the  gravity  of  the 
water  or  is  exceeded  by  it.  But  those  things  which  neither  by  preponder 
ating  descend,  nor,  by  yielding  to  the  preponderating  fluid,  ascend,  although 
by  their  true  weight  they  do  increase  the  weight  of  the  whole,  yet  com 
paratively,  and  in  the  sense  of  the  vulgar,  they  do  not  gravitate  in  the  wa 
ter.  For  these  cases  are  alike  demonstrated. 

COR.  7.  These  things  which  have  been  demonstrated  concerning  gravity 
take  place  in  any  other  centripetal  forces. 

COR.  8.  Therefore  if  the  medium  in  which  any  body  moves  be  acted  on 
either  by  its  own  gravity,  or  by  any  other  centripetal  force,  and  the  body 
be  urged  more  powerfully  by  the  same  force  ;  the  difference  of  the  forces  is 
that  very  motive  force,  which,  in  the  foregoing  Propositions,  I  have  con 
sidered  as  a  centripetal  force.  But  if  the  body  be  more  lightly  urged  by 
that  force,  the  difference  of  the  forces  becomes  a  centrifugal  force,  and  is  tc 
be  considered  as  such. 

COR.  9.  But  since  fluids  by  pressing  the  included  bodies  do  not 
change  their  external  figures,  it  appears  also  (by  Cor.  Prop.  XIX)  that  they 
will  not  change  the  situation  of  their  internal  parts  in  relation  to  onf 
another ;  and  therefore  if  animals  were  immersed  therein,  and  that  all  sen 
sation  did  arise  from  the  motion  of  their  parts,  the  fluid  will  neither  hurt 
the  immersed  bodies,  nor  excite  any  sensation,  unless  so  far  as  those  bodies 
may  be  condensed  by  the  compression.  And  the  case  is  the  same  of  any 
system  of  bodies  encompassed  with  a  compressing  fluid.  All  the  parts  of 
the  system  will  be  agitated  with  the  same  motions  as  if  they  were  placed 
in  a  vacuum,  and  would  only  retain  their  comparative  gravity  ;  unless  so 
far  as  the  fluid  may  somewhat  resist  their  motions,  or  be  requisite  to  con- 
glutinate  them  by  compression. 

PROPOSITION  XXI.     THEOREM  XVI. 

£<et  the  density  of  any  fluid  be  proportional  to  the  compression,  and  its 
parts  be  attracted  downwards  by  a  centripetal  force  reciprocally  pro 
portional  to  the  distances  from  the  centre :  I  say,  that,  if  those  dis 
tances  be  taken  continually  proportional,  the  densities  of  the  fluid  at 
the  same  distances  will  be  also  continually  proportional. 
Let  ATV  denote  the  spherical  bottom  of  the  fluid,  S  the  centre,  S  A,  SB. 
SC,  SD,  SE,  SF,  &c.,  distances  continually  proportional.     Erect  the  per 
pendiculars  AH,  BI,  CK,  DL,  EM,  PN,  &c.,  which  shall  be  as  the  densi 
ties  of  the  medium  in  the  places  A,  B,  C,  D,  E,  F  :  and  the  specific  grav 

ATT    RT  f^K" 
ities  in  those  places  will  be  aa  -r-,       ,  -       &c.,  or,  which  is  all  one,  a&- 


298 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK  II. 


AH    BI  CK 

ATT  BC'  CD'  Suppose,  first,  these  gravities  to  be  uniformly  continued 

from  A  to  B,  from  B  to  C,  from  C  to  D,  (fee.,  the  decrements  in  the  points 
B,  C,  D,  (fee.,  being  taken  by  steps.  Arid  these  gravi 
ties  drawn  into  the  altitudes  AB,  BC,  CD,  (fee.,  will 
give  the  pressures  AH,  BI,  CK,  (fee.,  by  which  the  bot 
tom  ATV  is  acted  on  (by  Theor.  XV).  Therefore  the 
particle  A  sustains  all  the  pressures  AH,  BI,  CK,  DJL, 
(fee.,  proceeding  in  infinitum  ;  and  the  particle  B  sus 
tains  the  pressures  of  all  but  the  first  AH  ;  and  the  par 
ticle  C  all  but  the  two  first  AH,  BI ;  and  so  on  :  and 
therefore  the  density  AH  of  the  first  particle  A  is  to 
the  density  BI  of  the  second  particle  B  as  the  sum  of 
all  AH  -f-  BI  +  CK  +  DL,  in  infinitum,  to  the  sum  of 
all  BI  -f-  CK  -f  DL,  (fee.  And  BI  the  density  of  the  second  particle  B  is 
to  CK  the  density  of  the  third  C,  as  the  sum  of  all  BI  -f  CK  +  DL,  (fee., 
to  the  sum  of  all  CK  -f-  DL,  (fee.  Therefore  these  sums  are  proportional 
to  their  differences  AH,  BI,  CK,  (fee.,  and  therefore  continually  propor 
tional  (by  Lem.  1  of  this  Book) ;  and  therefore  the  differences  AH,  BI, 
CK,  (fee.,  proportional  to  the  sums,  are  also  continually  proportional. 
Wherefore  since  the  densities  in  the  places  A,  B,  C,  (fee.,  are  as  AH,  BI, 
CK,  (fee.,  they  will  also  be  continually  proportional.  Proceed  intermis- 
sively,  and,  ex  ccquo,  at  the  distances  SA,  SC,  SE,  continially  proportional, 
the  densities  AH,  CK,  EM  will  be  continually  proportional.  And  by  the 
same  reasoning,  at  any  distances  SA,  SD,  SG,  continually  proportional, 
the  densities  AH.  DL,  GO,  will  be  continually  proportional.  Let  now  the 
points  A,  B,  C.  D,  E,  (fee.,  coincide,  so  that  the  progression  of  the  specif'.c 
gravities  from  the  bottom  A  to  the  top  of  the  fluid  may  be  made  continual ; 
and  at  any  distances  SA,  SD,  SG,  continually  proportional,  the  densities 
AH,  DL,  GO,  being  all  along  continually  proportional,  will  still  remain 
continually  proportional.  Q.E.D. 

COR.  Hence  if  the  density  of  the  fluid  in  two  places, 
as  A  and  E,  be  given,  its  density  in  any  other  place  Q, 
may  be  collected.  With  the  centre  S,  and  the  rectan 
gular  asymptotes  SQ,  SX,  describe  an  hyperbola  cut 
ting  the  perpendiculars  AH,  EM,  QT  in  «,  e,  and  q} 
as  also  the  perpendiculars  HX,  MY,  TZ,  let  fall  upon 
the  asypmtote  SX,  in  //,  'm,  and  t.  Make  the  area 
Y////Z  to  the  given  area  YmAX  as  the  given  area 
EeqQ  to  the  given  area  EeaA ;  and  the  line  Z£  produced  will  cut  off  the 
line  Q,T.  proportional  to  the  density.  For  if  the  lines  SA,  SE,  SQ  are 
continually  proportional,  the  areas  ReqQ.,  fyaA  will  be  equal,  and  thence 


X 


SEC.   V. 


OF    NATURAL    PHILOSOPHY. 


299 


the  areas  YwYZ.  X/zwY,  proportional  to  them,  will  be  also  equal ;  and 
the  lines  SX,  SY,  SZ,  that  is,  AH,  EM,  Q,T  continually  proportional,  as 
they  ought  to  be.  And  if  the  lines  SA,  SE,  SQ,5  obtain  any  other  order 
in  the  series  of  continued  proportionals,  the  lines  AH,  EM,  Q,T,  because 
of  the  proportional  hyperbolic  areas,  will  obtain  the  same  order  in  another 
series  of  quantities  continually  proportional. 


PROPOSITION  XXII     THEOREM  XVII. 

Let  the  density  of  any  fluid  be  proportional  to  the  compression,  and  its 
parts  be  attracted  downwards  by  a  gravitation  reciprocally  propor 
tional  to  the  squares  of  the  distances  from  the  centre  :  I  say,  that  if 
the  distances  be  taken  in  harmonic  progression,  the  densities  of  the 
fluid  at  those  distances  will  be  in  a  geometrical  progression. 
Let  S  denote  the  centre,  and  SA, 

SB,  SC,  SD,   SE,  the  distances  in 

geometrical  progression.     Erect  the 

perpendiculars   AH,    BI,  CK,   (fee., 

which  shall  be    as  the  densities   of  c 

the  fluid  in  the  places  A,  B,  C,  D,  » 

E,   (fee.,    and    the   specific   gravities 

thereof   in    those  places   will  be  as 


AH     BI 


,^-,  (fee.     Suppose  these 


V 

LN 

M 

I. 

K 

i 

V 

V 

V 

V7 

^ 

n 

w 
r 

/ 

£ 

L  / 

/ 

/t 

SA2'SB2'SC2' 

gravities  to  be  uniformly  continued,  the  first  from  A  to  B,  the  second  from 
B  to  C,  the  third  from  C  to  I),  &c.  And  these  drawn  into  the  altitudes 
AB,  BC,  CO,  DE,  (fec.;  or,  which  is  the  same  thing/into  the  distances  SA, 

ATT    r>T     OT7" 

SB,  SC,  (fee.,  proportional  to  those  altitudes,  will  give  -~-r-,  ^=5,  -~~,  (fee.. 

the  exponents  of  the  pressures.  Therefore  since  the  densities  are  as  th^ 
sums  of  those  pressures,  the  differences  AH  —  BI,  BI — CK,  (fee.,  of  tb,« 

densities  will  be  as  the  differences  of  those  sums  ~-r~,  ^,  ~~,   (fee.     With 

the  centre  S,  and  the  asymptotes  SA,  S#,  describe  any  hyperbola,  cutting 
the  perpendiculars  AH,  BI,  CK,  (fee.,  in  a,  6,  c,  (fee.,  and  the  perpendicu 
lars  H/,  I//,,  K?#,  let  fall  upon  the  asymptote  Sv,  in  h,  i,  k  ;  and  the  dif 
ferences  of  the  densities  tu,  uw,  (fee.,  will  be  as  „  A ,  ^^,  (fee.  And  the 


SA;  SB; 


rectangles  tu  X  th,  uw  X  uij  (fee.,  or  tp,  uq,  (fee.,  as 
that  is,  as  Aa,  Bb,  (fee. 


AH  X  th  BI  X  ui 

— ,  (fee. 


SA  SB 

For,  by  the  nature  of  the  hyperbola,  SA  is  to  AH 

or  St  as  th  to  Ar,  and  therefore  — pri —  is  equal  to  Aa .     And,  by  a  like 


SA 


300  THE    MATHEMATICAL    PRINC.  PLES  [BOOK    II. 

reasoning,  —  ^n~~  *s  e(lua^  to  ^,  &c-     But  Aa>  B^>  ^c,  &cv  are  continu 

ally  proportional,  and  therefore  proportional  to  their  differences  Aa  —  B&, 
B6  —  Cc;  &c.,  therefore  the  rectangles  fy?,  nq,  &c.,  are  proportional  to  those 
differences  ;  as  also  the  sums  of  the  rectangles  tp  +  uq,  or  tp  +  uq  -f  w 
to  the  sums  of  the  differences  Aa  —  Cc  or  Aa  —  Da7.  Suppose  several  of 
these  terms,  and  the  sum  of  all  the  differences,  as  Aa  —  F/,  will  be  pro 
portional  t?  the  sum  of  all  the  rectangles,  as  zthn.  Increase  the  number 
of  terms,  and  diminish  the  distances  of  the  points  A,  B,  C,  (fee.,  in  iiijini- 
tum,  and  those  rectangles  will  become  equal  to  the  hyperbolic  area  zthn. 
and  therefore  the  difference  Aa  —  F/  19  proportional  to  this  area.  Take 
nowT  any  distances,  as  SA,  SD,  SF,  in  harmonic  progression,  and  the  dif 
ferences  Aa  —  Da7,  Da1  —  F/  will  be  equal  ;  and  therefore  the  areas  thlx, 
xlnz,  proportional  to  those  differences  will  be  equal  among  themselves,  and 
the  densities  St,  S:r,  Sz,  that  is,  AH,  DL,  FN,  continually  proportional. 
Q.E.D. 

COR.  Hence  if  any  two  densities  of  the  fluid,  as  AH  and  BI,  be  given, 
the  area  thiu,  answering  to  their  difference  tu,  will  be  given;  and  thence 
the  density  FN  will  be  found  at  any  height  SF,  by  taking  the  area  thnz  to 
that  given  area  thiu  as  the  difference  Aa  —  F/  to  the  difference  Aa  —  Eh. 

SCHOLIUM. 

By  a  like  reasoning  it  may  be  proved,  that  if  the  gravity  of  the  particles 
of  a  fluid  be  diminished  in  a  triplicate  ratio  of  the  distances  from  the  centre  ; 
and  the  reciprocals  of  the  squares  of  the  distances  SA,  SB,  SC,  &c.,  (namely, 

SA3  SA3  SA3   . 

opt'  ^e  ta^en  m  an  arithmetical  progression,  the  densities  AH. 


BI,  CK,  &c.,  will  be  in  a  geometrical  progression.  And  if  the  gravity  be 
diminished  in  a  quadruplicate  ratio  of  the  distances,  and  the  reciprocals  of 

the  cubes  of  the  distances  (as  ^-r^,  SRS»  sps'  ^c'^  ^e  ta^cn  ^n  ai>itnmeti- 

cai  progression,  the  densities  AH,  BI,  CK,  &c.,  will  be  in  geometrical  pro 
gression.  And  so  in  irtfinitum.  Again  ;  if  the  gravity  of  the  particles  of 
the  fluid  be  the  same  at  all  distances,  and  the  distances  be  in  arithmetical 
progression,  the  densities  will  be  in  a  geometrical  progression  as  Dr.  Hal- 
ley  has  found.  If  the  gravity  be  as  the  distance,  and  the  squares  of  the 
distances  be  in  arithmetical  progression,  the  densities  will  be  in  geometri 
cal  progression.  And  so  in  infinitum.  These  things  will  be  so,  when  the 
density  of  the  fluid  condensed  by  compression  is  as  the  force  of  compres 
sion  ;  or,  which  is  the  same  thing,  when  the  space  possessed  by  the  fluid  is 
reciprocally  as  this  force.  Other  laws  of  condensation  may  be  supposed, 
as  that  the  cube  of  the  compressing  force  may  be  as  the  biquadrate  of  the 


SEC.    V.]  OF    NATURAL    PHILOSOPHY.  301 

de  isity  ;  or  the  triplicate  ratio  of  tlie  force  the  same  with  the  quadruplicate 
ratio  of  the  density  :  in  which  case,  if  the  gravity  he  reciprocally  as  the 
square  of  the  distance  from  the  centre,  the  density  will  be  reciprocally  at 
the  cube  of  the  distance.  Suppose  that  the  cube  of  the  compressing  force 
be  as  the  quadrato-cube  of  the  density ;  and  if  the  gravity  be  reciprocally 
as  the  square  of  the  distance,  the  density  will  be  reciprocally  in  a  sesqui- 
plicate  ratio  of  the  distance.  Suppose  the  compressing  force  to  be  in  a  du 
plicate  ratio  of  the  density,  and  the  gravity  reciprocally  in  a  duplicate  ra 
tio  of  the  distance,  and  the  density  will  be  reciprocally  as 'the  distance. 
To  run  over  all  the  cases  that  might  bo  offered  would  be  tedious.  But  as 
to  our  own  air,  this  is  certain  from  experiment,  that  its  density  is  either 
accurately,  or  very  nearly  at  least,  as  the  compressing  force ;  and  therefore 
the  density  of  the  air  in  the  atmosphere  of  the  earth  is  as  the  weight  of 
the  whole  incumbent  air,  that  is,  as  the  height  of  the  mercury  in  the  ba 
rometer. 

PROPOSITION  XXIII.     THEOREM  XVIII. 

If  a  fluid  be  composed  of  particles  mutually  flying  each  other,  and  the 
drnsity  be  as  the  compression,  the  centrifugal  forces  of  the  particles 
'will  be  reciprocally  proportional  to  tlie  distances  of  their  centres.     And, 
vice  versa,  particles  flying  each  otli,er,  with  forces  that  are  reciprocally 
proportional  to  the  distances  of  their  centres^  compose  an  elastic  fluid, 
whose  density  is  as  the  compression. 
Let  the  fluid  be  supposed  to  be  included  in  a  cubic 
space  ACE,  and  then  to  be  reduced  by  compression  into 
a  lesser  cubic  space  ace  ;  and  the  distances  of  the  par-  F 
tides  retaining  a  like  situation  with  respect  to  each 
other  in  both  the  spaces,  will  be  as  the  sides  AB,  ab  of 
the  cubes  ;  and  the  densities  of  the  mediums  will  be  re 
ciprocally  as  the  containing  spaces  AB3,  ab3.     In  the 
plane  side  of  the  greater  cube  A  BCD  take  the  square 
DP  equal  to  the  plane  side  db  of  the  lesser  cube:  and, 
by  the  supposition,  the  pressure  with  which  the  square 
DP  urges  the  inclosed  fluid  will  be  to  the  pressure  with 
which  that  square  db  urges  the  inclosed  fluid  as  the  densities  of  the  me 
diums  are  to  each  other,  that  is,  asa/>3  to  AB3.     But  the  pressure  with 
which  the  square  DB  urges  the  included  fluid  is  to  the  pressure  with  which 
the  square  DP  urges  the  same  fluid  as  the  square  DB  to  the  square  DP, 
that  is,  as  AB2  to  abz.     Therefore,  ex  cequo,  the  pressure  with  which  the 
square  DB  urges  the  fluid   is   to   the  pressure   with  which  the  square  db 
urges  the  fluid  as  ab  to  AB.     Let  the  planes  FGH,/°V?,  U  drawn  through 
the  middles  of  the  two  cubes,  and  divide  the  fluid  into  tw^/  parts,     These 
parts  will  press  each  other  mutually  with  the  same  forces  with  which  they 


A 


THE    MATHEMATICAL    PRINCIPLES  [BOOK    II. 

are  themselves  pressed  by  the  planes  AC,  ac,  that  is,  in  the  proportion  of 
ab  to  AB  :  arid  therefore  the  centrifugal  forces  by  which  these  pressures 
are  sustained  are  in  the  same  ratio.  The  number  of  the  particles  being 
equal,  and  the  situation  alike,  in  both  cubes,  the  forces  which  all  the  par 
ticles  exert,  according  to  the  planes  FGH,/o7/,,  upon  all,  are  as  the  forces 
which  each  exerts  on  each.  Therefore  the  forces  which  each  exerts  on 
each,  according  to  the  plane  FGH  in  the  greater  cube,  are  to  the  forces 
which  each  exerts  on  each,  according  to  the  plane  fgh  in  the  lesser  cube, 
us  ab  to  AB,*that  is,  reciprocally  as  the  distances  of  the  particles  from  each 
other.  Q.E.D. 

And,  vice  versa,  if  the  forces  of  the  single  particles  are  reciprocally  as 
the  distances,  that  is,  reciprocally  as  the  sides  of  the  cubes  AB,  ab  ;  the 
sums  of  the  forces  will  be  in  the  same  ratio,  and  the  pressures  of  the  sides 
i)B,  db  as  the  sums  of  the  forces ;  and  the  pressure  of  the  square  DP  to 
the  pressure  of  the  side  DB  as  ab2  to  AB 2.  And,  ex  cequo,  the  pressure  of 
the  square  DP  to  the  pressure  of  the  side  db  as  ab*  to  AB3  ;  that  is,  the 
force  of  compression  in  the  one  to  the  force  of  compression  in  the  other  as 
the  density  in  the  former  to  the  density  in  the  latter.  Q.E.D. 

SCHOLIUM. 

By  a  like  reasoning,  if  the  centrifugal  forces  of  the  particles  are  recip 
rocally  in  the  duplicate  ratio  of  the  distances  between  the  centres,  the  cubes 
of  the  compressing  forces  will  be  as  the  biquadrates  of  the  densities.  If 
the  centrifugal  forces  be  reciprocally  in  the  triplicate  or  quadruplicate  ratio 
of  the  distances,  the  cubes  of  the  compressing  forces  will  be  as  the  quadrato- 
cubes,  or  cubo-cubes  of  the  densities.  And  universally,  if  D  be  put  for  the 
distance,  and  E  for  the  density  of  the  compressed  fluid,  and  the  centrifugal 
forces  be  reciprocally  as  any  power  Dn  of  the  distance,  whose  index  is  the 
number  ??,  the  compressing  forces  will  be  as  the  cube  roots  of  the  power 
En  +  2.  whose  index  is  the  number  n  +  2  ;  and  the  contrary.  All  these 
things  are  to  be  understood  of  particles  whose  centrifugal  forces  terminate 
in  those  particles  that  are  next  them,  or  are  diffused  not  much  further. 
We  have  an  example  of  this  in  magnetical  bodies.  Their  attractive  vir 
tue  is  terminated  nearly  in  bodies  of  their  own  kind  that  are  next  them. 
The  virtue  of  the  magnet  is  contracted  by  the  interposition  of  an  iron 
plate,  and  is  almost  terminated  at  it :  for  bodies  further  off  are  not  attracted 
by  the  magnet  so  much  as  by  the  iron  plate.  If  in  this  manner  particles  repel 
others  of  their  own  kind  that  lie  next  them,  but  do  not  exert  their  virtue 
on  the  more  remote,  particles  of  this  kind  will  compose  such  fluids  as  are 
treated  of  in  this  Proposition,  If  the  virtue  of  any  particle  diffuse  itself 
every  way  in  inftnitum,  there  will  be  required  a  greater  force  to  produce 
an  equal  condensation  of  a  greater  quantity  of  the  flui  1.  But  whether 


SEC.   VI.]  OF    NATURAL    PHILOSOPHY.  303 

elastic  fluids  do  really  consist  of  particles  so  repelling  each  other,  is  a  phy 
sical  question.  We  have  here  demonstrated  mathematically  the  property 
of  fluids  consisting  of  particles  of  this  kind,  that  hence  philosophers  may 
take  occasion  to  discuss  that  question. 


SECTION  VI. 

Of  the  motion  and  resistance  of  funependulous  bodies. 

PROPOSITION  XXIV.     THEOREM  XIX. 

The  quantities  of  matter  i/i  funependulous  bodies,  whose  centres  of  oscil 
lation  are  equally  distant  from,  the  centre  of  suspension,  are  in  a,  ratio 
compounded  of  the  ratio  of  the  weights  and  the  duplicate  ratio  of  the 
times  of  the  oscillations  in  vacuo. 

For  the  velocity  which  a  given  force  can  generate  in  a  given  matter  in 
a  given  time  is  as  the  force  and  the  time  directly,  and  the  matter  inversely. 
The  greater  the  force  or  the  time  is,  or  the  less  the  matter,  the  greater  ve 
locity  will  he  generated.  This  is  manifest  from  the  second  Law  of  Mo 
tion.  Now  if  pendulums  are  of  the  same  length,  the  motive  forces  in  places 
equally  distant  from  the  perpendicular  are  as  the  weights  :  and  therefore 
if  two  bodies  by  oscillating  describe  equal  arcs,  and  those  arcs  are  divided 
into  equal  parts ;  since  the  times  in  which  the  bodies  describe  each  of  the 
correspondent  parts  of  the  arcs  are  as  the  times  of  the  whole  oscillations, 
the  velocities  in  the  correspondent  parts  of  the  oscillations  will  be  to  each 
other  as  the  motive  forces  and  the  whole  times  of  the  oscillations  directly, 
and  the  quantities  of  matter  reciprocally  :  and  therefore  the  quantities  of 
matter  are  as  the  forces  and  the  times  of  the  oscillations  directly  and  the 
velocities  reciprocally.  But  the  velocities  reciprocally  are  as  the  times, 
and  therefore  the  times  directly  and  the  velocities  reciprocally  are  as  the 
squares  of  the  times;  and  therefore  the  quantities  of  matter  are  as  the  mo 
tive  forces  and  the  squares  of  the  times,  that  is,  as  the  weights  and  the 
squares  of  the  times.  Q.E.D. 

COR.  1.  Therefore  if  the  times  are  equal,  the  quantities  of  matter  in 
each  of  the  bodies  are  as  the  weights. 

COR.  2.  If  the  weights  are  equal,  the  quantities  of  matter  will  be  as  the 
pquarcs  of  the  times. 

COR.  3.  If  the  quantities  of  matter  are  equal,  the  weights  will  be  recip 
rocally  as  the  squares  of  the  times. 

COR.  4.  Whence  since  the  squares  of  the  times,  cceteris  paribus,  are  as 
the  length*  of  the  pendulums,  therefore  if  both  the  times  and  quantities  of 
matter  are  equal,  the  weights  will  be  as  the  lengths  of  the  pendulums. 


J04  THE    MATHEMATICAL    PRINCIPLES  [BOOK    11 

COR.  5.  And  universally,  the  quantity  of  matter  in  the  pendulous  body 
is  as  the  weight  and  the  square  of  the  time  directly,  and  the  length  of  the 
pendulum  inversely. 

COR.  6.  But  in  a  non-resisting  medium,  the  quantity  of  matter  in  the 
pendulous  body  is  as  the  comparative  weight  and  the  square  of  the  time 
directly,  and  the  length  of  the  pendulum  inversely.  For  the  comparative 
weight  is  the  motive  force  of  the  body  in  any  heavy  medium,  as  was  shewn 
above  ;  and  therefore  does  the  same  thing  in  such  a  non-resisting  medium 
as  the  absolute  weight  does  in  a  vacuum. 

COR.  7.  And  hence  appears  a  method  both  of  comparing  bodies  one 
among  another,  as  to  the  quantity  of  matter  in  each  ;  and  of  comparing 
the  weights  of  the  same  body  in  different  places,  to  know  the  variation  of 
its  gravity.  And  by  experiments  made  with  the  greatest  accuracy,  I 
have  always  found  the  quantity  of  matter  in  bodies  to  be  proportional  to 
their  weight. 

PROPOSITION  XXV.     THEOREM  XX. 

Funependulous  bodies  that  are,  in,  any  'medium,  resisted  in  the  ratio  oj 
the  moments  of  time,  and  funepetidulons  bodies  that  move  in  a  non- 
resisting  medium  of  the  same  specific  gravity,  perform  their  oscilla 
tions  in.  a  cycloid  in  the  same  time,  and  describe  proportional  parts  oj 
arcs  together. 

Let  AB  be  an  arc  of  a  cycloid,  which 
a  body  D,  by  vibrating  in  a  non-re 
sisting  medium,  shall  describe  in  any 
time.  Bisect  that  arc  in  C,  so  that  C 
may  be  the  lowest  point  thereof  ;  and 
the  accelerative  force  with  which  the 
body  is  urged  in  any  place  D,  or  d  or 
E,  will  be  as  the  length  of  the  arc  CD, 


pressed  by  that  same  arc  ;  and  since  the  resistance  is  as  the  moment  of  the 
time,  and  therefore  given,  let  it  ba  expressed  by  the  given  part  CO  of  the 
cycloidal  arc,  and  take  the  arc  Od  in  the  same  ratio  to  the  arc  CD  that 
the  arc  OB  has  to  the  arc  CB  :  and  the  force  with  which  the  body  in  d  is 
urged  in  a  resisting  medium,  being  the  excess  of  the  force  Cd  above  the 
resistance  CO,  will  be  expressed  by  the  arc  Od,  and  will  therefore  be  to 
the  force  with  which  the  body  D  is  urged  in  a  non-resisting  medium  in  the 
place  D,  as  the  arc  Od  to  the  arc  CD  ;  and  therefore  also  in  the  place  B, 
as  the  arc  OB  to  the  arc  CB.  Therefore  if  two  bodies  D,  d  go  from  the  place 
B,  and  are  urged  by  these  forces  ;  since  the  forces  at  the  beginning  are  as 
the  arc  CB  and  OB,  the  first  velocities  and  arcs  first  described  will  be  in 
the  same  ratio.  Let  those  arcs  be  BD  and  Ed,  and  the  remaining  arcf 


SEC.    VI. |  OF    NATURAL    PHILOSOPHY.  305 

CD,  Odj  will  be  in  the  same  ratio.  Therefore  the  forces,  being  propor 
tional  to  those  arcs  CD,  Od,  will  remain  in  the  same  ratio  as  at  the  be 
ginning,  and  therefore  the  bodies  will  continue  describing  together  arcs  in 
the  same  ratio.  Therefore  the  forces  and  velocities  and  the  remaining  arcs 
CD.  Od,  will  be  always  as  the  whole  arcs  CB,  OB,  and  therefore  those  re 
maining  arcs  wLl  be  described  together.  Therefore  the  two  bodies  D  and 
d  will  arrive  together  at  the  places  C  and  O  ;  that  whicli  moves  in  the 
non-resisting  medium,  at  the  place  C,  and  the  other,  in  the  resisting  me 
dium,  at  the  place  O.  Now  since  the  velocities  in  C  and  O  are  as  the  arcs 
CB,  OB,  the  arcs  which  the  bodies  describe  when  they  go  farther  will  be 
in  the  same  ratio.  Let  those  arcs  be  CE  and  Oe.  The  force  with  which 
the  body  D  in  a  non-resisting  medium  is  retarded  in  E  is  as  CE,  and  the 
force  with  which  the  body  d  in  the  resisting  medium  is  retarded  in  e,  is  as 
the  sum  of  the  force  Ce  and  the  resistance  CO,  that  is,  as  Oe  ;  and  there 
fore  the  forces  with  which  the  bodies  are  retarded  are  as  the  arcs  CB,  OB, 
proportional  to  the  arcs  CE,  Oe  ;  and  therefore  the  velocities,  retarded  in 
that  given  ratio,  remain  in  the  same  given  ratio.  Therefore  the  velocities 
and  the  arcs  described  with  those  velocities  are  always  to  each  other  in 
that  given  ratio  of  the  arcs  CB  and  OB  ;  and  therefore  if  the  entire  arcs 
AB,  aB  are  taken  in  the  same  ratio,  the  bodies  D  andc/  will  describe  those 
aics  together,  and  in  the  places  A  and  a  will  lose  all  their  motion  together. 
Therefore  the  whole  oscillations  are  isochronal,  or  are  performed  in  equal 
times  ;  and  any  parts  of  the  arcs,  as  BD,  Ed,  or  BE,  Be,  that  are  described 
together,  are  proportional  to  the  whole  arcs  BA,  B«.  Q,.E.D. 

COR.  Therefore  the  swiftest  motion  in  a  resisting  medium  does  not  fall 
upon  the  lowest  point  C,  but  is  found  in  that  point  O,  in  which  the  whole 
arc  described  Ba  is  bisected.  And  the  body,  proceeding  from  thence  to  a, 
is  retarded  at  the  same  rate  with  which  it  was  accelerated  before  in  its  de 
scent  from  B  to  O. 

PROPOSITION  XXVI.    THEOREM  XXI. 

Funependulous  bodies,  that  are  resisted  in  the  ratio  of  the  velocity,  have 

their  oscillations  in  a  cycloid  isochronal. 

For  if  two  bodies,  equally  distant  from  their  centres  of  suspension,  de 
scribe,  in  oscillating,  unequal  arcs,  and  the  velocities  in  the  correspondent 
parts  of  the  arcs  be  to  each  other  as  the  whole  arcs ;  the  resistances,  pro 
portional  to  the  velocities,  will  be  also  to  each  other  as  the  same  arcs. 
Therefore  if  these  resistances  be  subducted  from  or  added  to  the  motive 
forces  arising  from  gravity  which  are  as  the  same  arcs,  the  differences  or 
sums  will  be  to  each  other  in  the  same  ratio  of  the  arcs ;  and  since  the  in 
crements  and  decrements  of  the  velocities  are  as  these  differences  or  sums, 
the  velocities  will  be  always  as  the  whole  arcs;  therefore  if  the  velocities 
are  in  any  one  case  as  the  whole  arcs,  they  will  remain  always  in  the  same 

20 


306  THE    MATHEMATICAL    PRINCIPLES  [BOOK.    11 

ratio.  But  at  the  beginning  of  the  motion,  when  the  bodies  begin  to  de 
scend  and  describe  those  arcs,  the  forces,  which  at  that  time  are  proportional 
to  the  arcs,  will  generate  velocities  proportional  to  the  arcs.  Therefore 
the  velocities  will  be  always  as  the  whole  arcs  to  be  described,  and  there 
fore  those  arcs  will  be  described  in  the  same  time.  Q,.E.D. 


PROPOSITION  XXVII.     THEOREM  XXII. 

If  fnnependulous  bodies  are  resisted  in  the  duplicate  ratio  of  their 
velocities,  the  differences  between  the  times  of  the  oscillations  in  a  re 
sisting  medium,  and  the  times  of  the  oscillations  in  a  non-resisting 
medium  of  the  same  specific  gravity,  will  be  proportional  to  the  arcs 
described  in  oscillating  nearly. 

For  let  equal  pendulums  in  a  re 
sisting  medium  describe  the  unequal 
arcs  A,  B  ;  and  the  resistance  of  the 
body  in  the  arc  A  will  be  to  the  resist 
ance  of  the  body  in  the  correspondent 
part  of  the  arc  B  in  the  duplicate  ra 
tio  of  the  velocities,  that  is,  as,  A  A  to 
BB  nearly.  If  the  resistance  in  the 
arc  B  were  to  the  resistance  in  the  arc 
A  as  AB  to  AA,  the  times  in  the  arcs  A  and  B  would  be  equal  (by  the  last 
Prop.)  Therefore  the  resistance  AA  in  the  arc  A,  or  AB  in  the  arc  B, 
causes  the  excess  of  the  time  in  the  arc  A  above  the  time  in  a  non-resisting 
medium ;  and  the  resistance  BB  causes  the  excess  of  the  time  in  the  arc  B 
above  the  time  in  a  non-resisting  medium.  But  those  excesses  are  as  the 
efficient  forces  AB  and  BB  nearly,  that  is,  as  the  arcs  A  and  B.  Q.E.D. 

COR,  1.  Hence  from  the  times  of  the  oscillations  in  unequal  arcs  in  a 
resisting  medium,  may  be  knowrn  the  times  of  the  oscillations  in  a  non- re 
sisting  medium  of  the  same  specific  gravity.  For  the  difference  of  the 
times  will  be  to  the  excess  of  the  time  in  the  lesser  arc  above  the  time  in  a 
non-resisting  medium  as  the  difference  of  the  arcs  to  the  lesser  arc. 

COR.  2.  The  shorter  oscillations  are  more  isochronal,  and  very  short 
ones  are  performed  nearly  in  the  same  times  as  in  a  non-resisting  medium. 
But  the  times  of  those  which  are  performed  in  greater  arcs  are  a  little 
greater,  because  the  resistance  in  the  descent  of  the  body,  by  which  the 
time  is  prolonged,  is  greater,  in  proportion  to  the  length  described  in  the 
descent  than  the  resistance  in  the  subsequent  ascent,  by  which  the  time  is 
contracted.  But  the  time  of  the  oscillations,  both  short  arid  long,  seems  to 
be  prolonged  in  some  measure  by  the  motion  of  the  medium.  For  retard 
ed  bodies  are  resisted  somewhat  less  in  proportion  to  the  velocity,  and  ac 
celerated  bodies  somewhat  more  than  those  that  proceed  uniformly  forwards ; 


SEC.  VI.] 


OF    NATURAL    PHILOSOPHY. 


307 


because  the  medium,  by  the  motion  it  has  received  from  the  bodies,  going 
forwards  the  same  way  with  them,  is  more  agitated  in  the  former  case,  and 
less  in  the  latter ;  and  so  conspires  more  or  less  with  the  bodies  moved. 
Therefore  it  resists  the  pendulums  in  their  descent  more,  and  in  their  as 
cent  less,  than  in  proportion  to  the  velocity;  and  these  two  causes  concur 
ring  prolong  the  time. 

PROPOSITION  XXVIII.     THEOREM  XXIII. 

If  afunependvlous  body,  oscillating  in  a  cycloid,  be  resisted  in  the  rati  > 
of  the  moments  of  the  time,  its  resistance  will  be  to  the  force  of  grav 
ity  as  the  excess  of  the  arc  described  in  the  whole  descent  above  the 
arc  described  in  the  subsequent  ascent  to  twice  the  length  of  the  pen 
dulum. 

Let  BC  represent  the  arc  described 
in  the  descent,  Ca  the  arc  described  in 
the  ascent,  and  Aa  the  difference  of 
the  arcs  :  and  things  remaining  as  they 
were  constructed  and  demonstrated  in 
Prop.  XXV,  the  force  with  which  the 
oscillating  body  is  urged  in  any  place 
D  will  be  to  the  force  of  resistance  as 
the  arc  CD  to  the  arc  CO,  which  is 
half  of  that  difference  Aa.  Therefore  the  force  with  which  the  oscillating 
body  is  urged  at  the  beginning  or  the  highest  point  of  the  cycloid,  that  is, 
the  force  of  gravity,  will  be  to  the  resistance  as  the  arc  of  the  cycloid,  be 
tween  that  highest  point  and  lowest  point  C,  is  to  the  arc  CO ;  that  is 
(doubling  those  arcs),  as  the  whole  cycloidal  arc,  or  twice  the  length  of  the 
pendulum,  to  the  arc  Aa.  Q.E.D. 

PROPOSITION  XXIX.     PROBLEM  VI. 

Supposing  that  a  body  oscillating  in  a  cycloid  is  resisted  in  a  duplicate 

ratio  of  the  velocity:  to  find  the  resistance  in  each  place. 
Let  Ba  be  an  arc  described  in  one  entire  oscillation,  C  the  lowest  point 


C    O 


K 


O      ,S  P      rR  Q        M 

of  the  cycloid,  and  CZ  half  the  whole  cycloidal  arc,  equal  to  the  length  of 
the  pendulum  ;  and  let  it  be  required  to  find  the  resistance  of  the  body  is 


30S  THE    MATHEMATICAL    PRINCIPLES  [BOOK   1L 

any  place  D.  Cut  the  indefinite  right  line  OQ  in  the  points  O,  S,  P,  Q,, 
so  that  (erecting  the  perpendiculars  OK,  ST,  PI,  QE,  and  with  the  centre 
O,  and  the  aysmptotcs  OK,  OQ,  describing  the  hyperbola  TIGE  cutting 
the  perpendiculars  ST,  PI,  QE  in  T.  I,  and  E,  and  through  the  point  I 
drawing  KF.  parallel  to  the  asymptote  OQ,  meeting  the  asymptote  OK  i  i 
K,  and  the  perpendiculars  ST  and  QE  in  L  and  F)  the  hyperbolic  area 
PIEQ  may  be  to  the  hyperbolic  area  PITS  as  the  arc  BC,  described  in  the 
descent  of  the  body,  to  the  arc  Ca  described  in  the  ascent  ;  and  that  the 
area  IEF  may  be  to  the  area  ILT  as  OQ  to  OS.  Then  with  the  perpen 
dicular  MN  cut  off  the  hyperbolic  area  PINM,  and  let  that  area  be  to  the 
hyperbolic  area  PIEQ  as  the  arc  CZ  to  the  arc  BC  described  in  the  de 
scent.  And  if  the  perpendicular  RG  cut  off  the  hyperbolic  area  PIGR, 
which  shall  be  to  the  area  PIEQ  as  any  arc  CD  to  the  arc  BC  described 
in  the  whole  descent,  the  resistance  in  any  place  D  will  be  to  the  force  of 
OR 

gravity  as  the  area          IEF  —  IGH  to  the  area  PINM. 


For  since  the  forces  arising  from  gravity  with  which  the  body  is 
urged  in  the  places  Z,  B,  D,  a,  are  as  the  arcs  CZ.  CB,  CD,  Ca  and  those 
arcs  are  as  the  areas  PINM,  PIEQ,  PIGR,  PITS;  let  those  areas  be  the 
exponents  both  of  the  arcs  and  of  the  forces  respectively.  Let  DC?  be  a 
very  small  space  described  by  the  body  in  its  descent  :  and  let  it  be  expressed 
r>y  the  very  small  area  RGor  comprehended  between  the  parallels  RG,  rg  ; 
and  produce  r<?  to  //,  so  that  GYlhg-  and  RG«r  may  be  the  contemporane 
ous  decrements  of  the  areas  IGH,  PIGR.  And  the  increment  Gllhg  — 

IEF,  or  Rr  X  HG  —  -^  IEF,  of  the  area  ~  IEF  —IGH  will  be 


, 
OQ  OQ 

IFF 

to  the  decrement  RG°r,  or  Rr  X  RG,  of  the  area  PIGR,  as  HG  —  -    - 


OR 

to  RG  ;  and  therefore  as  OR  X  HG  —         IEF  to  OR  X  GR  or  OP  X 


PL  that  is  (because  of  the  equal  quantities  OR  X  HG,  OR  X  HR  —  OR 
X  GR,  ORHK  —  OPIK,  PIHR  and  PIGR  +  IGH),  as  PIGR  +  IGH  — 
OR  OR 

IEF  to  OPIK.     Therefore  if  the  area  -        IEF  —  IGH  be  called 


OQ 

Y,  and  RGgr  the  decrement  of  the  area  PIGR  be  given,  the  increment  of 

the  area  Y  will  be  as  PIGR  —  Y. 

Then  if  V  represent  the  force  arising  from  the  gravity,  proportional  to 
the  arc  CD  to  be  described,  by  which  the  body  is  acted  upon  in  D,  and  R 
be  put  for  the  resistance,  V  —  R  will  be  the  whole  force  with  which  the 
body  is  urged  in  D.  Therefore  the  increment  of  the  velocity  is  as  V  —  R 
and  the  particle  of  time  in  which  it  is  generated  conjunctly.  But  the  ve 
locity  itself  is  as  the  contempo]  aueous  increment  of  the  space  described  di- 


SEC.  VI.J 


OF    NATURAL    PHILOSOPHY. 


309 


rectly  and  the  same  particle  of  time  inversely.  Therefore,  since  the  re 
sistance  is,  by  the  supposition,  as  the  square  of  the  velocity,  the  increment 
of  the  resistance  will  (by  Lem.  II)  be  as  the  velocity  and  the  increment  of 
the  velocity  conjunctly,  that  is,  as  the  moment  of  the  space  and  V  —  R 
conjunctly  ;  and,  therefore,  if  the  moment  of  the  space  be  given,  as  V  — 
11 ;  that  is,  if  for  the  force  V  we  put  its  exponent  PIGR,  and  the  resist 
ance  R  be  expressed  by  any  other  area  Z;  as  PIGR  —  Z.v 

Therefore  the  area  PIGR  uniformly  decreasing  by  the  subduction  of 
given  moments,  the  area  Y  increases  in  proportion  of  PIGR  —  Y,  and 
the  area  Z  in  proportion  of  PIGR  —  Z.  And  therefore  if  the  areas 
Y  and  Z  begin  together,  and  at  the  beginning  are  equal,  these,  by  the 
addition  of  equal  moments,  will  continue  to  be  equal  •  and  in  like  man 
ner  decreasing  by  equal  moments,  \vill  vanish  together.  And,  vice  versa, 
if  they  together  begin  and  vanish,  they  will  have  equal  moments  and  te 
always  equal ;  and  that,  because  if  the  resistance  Z  be  augmented,  the  ve 
locity  together  with  the  arc  C«,  described  in  the  ascent  of  the  body,  will  be 
diminished ;  and  the  point  in  which  all  the  motion  together  with  the  re 
sistance  ceases  coming  nearer  to  the  point  C,  the  resistance  vanishes  sooner 
than  the  area  Y.  And  the  contrary  will  happen  when  the  resistance  is 
diminished. 

Now  the  area  Z  begins  and  ends  where  the  resistance  is  nothing,  that  is, 
at  the  beginning  of  the  motion  where  the  arc  CD  is  equal  to  the  arc  CB, 


K 


/IK 


O       S    P       /~R    Q          M 

and  the  right  line  RG  falls  upon  the  right  line  Q.E  ;  and  at  the  end  of 
the  motion  where  the  arc  CD  is  equal  to  the  arc  Ca,  and  RG  falls  upon 

the  right  line  ST.     And  the  area*  Y  or  —  IEF  —  IGH  begins  and  ends 

also  where  the  resistance  is  nothing,  and  therefore  where  IEF     and 

IGH  are  equal  ;  that  is  (by  the  construction),  where  the  right  line  RG 
falls  successively  upon  the  right  lines  Q,E  and  ST.  Therefore  those  areas 
begin  and  vanish  together,  and  are  therefore  always  equal.  Therefore  the  area 
OR 

IEF  —  IGH  is  equal  to  the  area  Z,  by  which  the  resistance  is  ex 


pressed,  and  therefore  is  to  the  area  PINM,  by  which  the  gravity  is  ex 
pressed,  as  the  resistance  to  the  gravity.     Q.E.D. 


310 


THE    MATHEMATICAL    PRINCIPLES 


[BOOK    11. 


COR.  1 .  Therefore  the  resistance  in  the  lowest  place  C  is  to  the  force 

OP 

of  gravity  as  the  area  ^  ~  IEF  to  the  area  PINM. 

COR.  2.  But  it  becomes  greatest  where  the  area  PIHR  is  to  the  area 
IEF  as  OR  to  OQ.  For  in  that  case  its  moment  (that  is,  PIGR  —  Y) 
becomes  nothing. 

COR.  3.  Hence  also  may  be  known  the  velocity  in  each  place,  as  being 
in  the  subduplicate  ratio  of  the  resistance,  and  at  the  beginning  of  the  mo 
tion  equal  to  the  velocity  of  the  body  oscillating  in  the  same  cycloid  with 
out  any  resistance. 

However,  by  reason  of  the  difficulty  of  the  calculation  by  which  the  re 
sistance  and  the  velocity  are  found  by  this  Proposition,  we  have  thought 
fit  to  subjoin  the  Proposition  following. 

PROPOSITION  XXX.     THEOREM  XXIV. 

If  a  right  line  aB  be  equal  to  the  arc  of  a  cycloid  which  an  oscillating 
body  describes,  and  at  each  of  its  points  D  the  perpendiculars  DK  be 
erected,  which  shall  be  to  the  length  of  the  pendulum  as  the  resistance 
of  the  body  in  the  corresponding  points  of  the  arc  to  the  force  of  grav 
ity  ;  I  say,  that  the  difference  between  the  arc  described  in  the  whole 
descent  and  the  arc  described  in  the  whole  subsequent  ascent  drawn 
into  half  the  sum  of  the  same  arcs  will  be  equal  to  the  area  BKa 
which  all  those  perpendiculars  take  up. 

Let  the  arc  of  the  cycloid,  de 
scribed  in  one  entire  oscillation,  be 
expressed  by  the  right  line  aB, 
equal  to  it,  and  the  arc  which 
would  have  been  described  in  vaciw 
by  the  length  AB.  Bisect  AB  in 
C,  and  the  point  C  will  represent 
the  lowest  point  of  the  cycloid,  and 
CD  Mill  be  as  the  force  arising  from  gravity,  with  which  the  body  in  D  i,s 
urged  in  the  direction  of  the  tangent  of  the  cycloid,  and  will  have  the  same 
ratio  to  the  length  of  the  pendulum  as  the  force  in  D  has  to  the  force  of 
gravity.  Let  that  force,  therefore,  be  expressed  by  that  length  CD,  and 
the  force  of  gravity  by  the  length  of  the  pendulum ;  and  if  in  DE  you 
take  DK  in  the  same  ratio  to  the  length  of  the  pendulum  as  the  resistance 
has  to  the  gravity,  DK  will  be  the  exponent  of  the  resistance.  From  the 
centre  C  with  the  interval  CA  or  CB  describe  a  semi-circle  BEeA.  Let 
the  body  describe,  in  the  least  time,  the  space  Dd  ;  and,  erecting  the  per 
pendiculars  DE,  de,  meeting  the  circumference  in  E  and  e,  they  will  be  as 
the  velocities  which  the  body  descending  in  vacuo  from  the  point  B  would 
acquire  in  the  places  D  and  d.  This  appears  by  Prop,  LII,  Book  L  Let 


SEC.  VLJ  OF    NATURAL    PHILOSOPHY.  311 

therefore,  these  velocities  be  expressed  by  those  perpendiculars  DE,  de  ; 
arid  let  DF  be  the  velocity  which  it  acquires  in  D  by  falling  from  B  in 
the  resisting  medium.  And  if  from  the  centre  C  with  the  interval  OF  we 
describe  the  circle  F/M  meeting  the  right  lines  de  and  AB  in  /  and  M, 
then  M  will  be  the  place  to  which  it  would  thenceforward,  without  farther 
resistance,  ascend,  and  (//"the  velocity  it  would  acquire  in  d.  Whence, 
also,  if  FO-  represent  the  moment  of  the  velocity  which  the  body  D,  in  de 
scribing  the  least  space  DC/,  loses  by  the  resistance  of  the  medium ;  and 
CN  be  taken  equal  to  Cg ;  then  will  N  be  the  place  to  which  the  body,  if 
it  met  no  farther  resistance,  would  thenceforward  ascend,  and  MN  will  be 
the  decrement  of  the  ascent  arising  from  the  loss  of  that  velocity.  Draw 
F/n  perpendicular  to  dft  and  the  decrement  Fg  of  the  velocity  DF  gener 
ated  by  the  resistance  DK  will  be  to  the  increment  fm  of  the  same  velo 
city,  generated  by  the  force  CD,  as  the  generating  force  DK  to  the  gener 
ating  force  CD.  But  because  of  the  similar  triangles  F////,  Fhg,  FDC, 
fm  is  to  Fm  or  Dd  as  CD  to  DF  ;  and,  ex  ceqno,  Fg  to  Dd  as  DK  to 
DF.  Also  Fh  is  to  Fg  as  DF  to  CF  ;  and,  ex  ax/uo  perturbate,  Fh  or 
MN  to  Do1  as  DK  to  CF  or  CM  ;  and  therefore  the  sum  of  all  the  MN  X 
CM  will  be  equal  to  the  sum  of  all  the  Dd  X  DK.  At  the  moveable 
point  M  suppose  always  a  rectangular  ordinate  erected  equal  to  the  inde 
terminate  CM,  which  by  a  continual  motion  is  drawn  into  the  whole 
length  Aa  ;  and  the  trapezium  described  by  that  motion,  or  its  equal,  the 
rectangle  Aa  X  |aB,  will  be  equal  to  the  sum  of  all  the  MN  X  CM,  and 
therefore  to  the  sum  of  all  the  Dd  X  DK,  that  is,  to  the  area  BKVTa 

O.E.D. 

COR.  Hence  from  the  law  of  resistance,  and  the  difference  Aa  of  the 
arcs  Ca}  CB,  may  be  collected  the  proportion  of  the  resistance  to  the  grav 
ity  nearly. 

For  if  the  resistance  DK  be  uniform,  the  figure  BKTa  will  be  a  rec 
tangle  under  Ba  and  DK;  and  thence  the  rectangle  under  ^Ba  and  Aa 
will  be  equal  to  the  rectangle  under  Ba  and  DK,  and  DK  will  be  equal  to 
jAa.  Wherefore  since  DK  is  the  exponent  of  the  resistance,  and  the 
length  of  the  pendulum  the  exponent  of  the  gravity,  the  resistance  will  be 
to  the  gravity  as  \Aa  to  the  length  of  the  pendulum  ;  altogether  as  in 
Prop.  XXVIII  is  demonstrated. 

If  the  resistance  be  as  the  velocity,  the  figure  BKTa  will  be  nearly  an 
ellipsis.  For  if  a  body,  in  a  non-resisting  medium,  by  one  entire  oscilla 
tion,  should  describe  the  length  BA,  the  velocity  in  any  place  D  would  be 
as  the  ordinate  DE  of  the  circle  described  on  the  diameter  AB.  There 
fore  since  Ea  in  the  resisting  medium,  and  BA  in  the  non-resisting  one, 
are  described  nearly  in  the  same  times ;  and  therefore  the  velocities  in  each 
of  the  points  of  Ba  are  to  the  velocities  in  the  correspondent  points  of  the 
length  BA.  nearly  as  Ba  is  to  BA ,  the  velocity  in  the  point  D  in  the  re- 


312 


THE    MATHEMATICAL    PRINCIPLES 


[BJOK    11. 


sisting  medium  will  be  as  the  ordinate  of  the  circle  or  ellipsis  described 
upon  the  diameter  Ba  ;  and  therefore  the  figure  BKVTa  will  be  nearly  ac 
ellipsis.  Since  the  resistance  is  supposed  proportional  to  the  velocity,  le\ 
OV  be  the  exponent  of  the  resistance  in  the  middle  point  O  ;  and  an  ellip 
sis  BRVSa  described  with  the  centre  O,  and  the  semi-axes  OB,  OV,  will 
be  nearly  equal  to  the  figure  BKVTa,  and  to  its  equal  the  rectangle  Act 
X  BO.  Therefore  Aa  X  BO  is  to  OV  X  BO  as  the  area  of  this  ellipsis 
to  OV  X  BO;  that  is,  Aa  is  to  OV  as  the  area  of  the  semi-circle  to  the 
square  of  the  radius,  or  as  1 1  to  7  nearly  ;  and,  therefore,  T7T Aa  is  to  the 
length  of  the  pendulum  as  the  resistance  of  the  oscillating  body  in  O  to 
its  gravity. 

Now  if  the  resistance  DK  be  in  the  duplicate  ratio  of  the  velocity,  the 
figure  BKVTa  will  be  almost  a  parabola  having  V  for  its  vertex  arid  OV 
for  its  axis,  and  therefore  will  be  nearly  equal  to  the  rectangle  under  f  Ba 
and  OV.  Therefore  the  rectangle  under  |Ba  and  Aa  is  equal  to  the  rec 
tangle  f  Ba  X  OV,  and  therefore  OV  is  equal  to  f  Aa  ;  and  therefore  the 
resistance  in  O  made  to  the  oscillating  body  is  to  its  gravity  as  f  Aa  to  the 
length  of  the  pendulum. 

And  I  take  these  conclusions  to  be  accurate  enough  for  practical  uses. 
For  since  an  ellipsis  or  parabola  BRVSa  falls  in  with  the  figure  BKVTa 
in  the  middle  point  V,  that  figure,  if  greater  towards  the  part  BRV  or 
VSa  than  the  other,  is  less  towards  the  contrary  part,  and  is  therefore 
nearly  equal  to  it. 

PROPOSITION  XXXI.     THEOREM  XXV. 

If  the  resistance  made  to  an  oscillating  body  in  each  of  the  proportional 
parts  of  the  arcs  described  be  augmented  or  diminished  in,  a  given  ra 
tio,  the  difference  between  the  arc  described  in  the  descent  and  the  arc 
described  in  the  subsequent  ascent  ivill  be  augmented  or  diminished  in 
the  same  ratio. 

For  that  difference  arises  from 
the  retardation  of  the  pendulum 
by  the  resistance  of  the  medium, 
and  therefore  is  as  the  whole  re 
tardation  and  the  retarding  resist 
ance  proportional  thereto.  In  the 
foregoing  Proposition  the  rectan- 

M  isr  u  c    o      «.-/  n  P  gle  under  the  right  line   ^aB  and 

the  difference  Aa  of  the  arcs  CB,  Ca,  was  equal  to  the  area  BKTa,  And 
that  area,  if  the  length  aB  remains,  is  augmented  or  diminished  in  the  ra 
tio  of  the  ordinates  DK  ;  that  is,  in  the  ratio  of  the  resistance  and  is  there 
fore  as  the  length  aB  and  the  resistance  conjunctly.  And  therefore  the 
rectangle  under  Aa  and  |aB  is  as  aB  and  the  resistance  conjunctly,  anc 
therefore  Aa  is  as  the  resistance.  QJE.D. 


SEC.    VI.l  OF    NATURAL    PHILOSOPHY.  313 

COR.  1.  Hence  if  the  resistance  be  as  the  velocity,  the  difference  of 
the  arts  in  the  same  medium  will  be  as  the  whole  arc  described :  and  the 
contrary. 

COR.  2.  If  the  resistance  be  in  the  duplicate  ratio  of  the  velocity,  that 
difference  will  be  in  the  duplicate  ratio  of  the  whole  arc  :  and  the  contrary. 

COR.  3.  And  universally,  if  the  resistance  be  in  the  triplicate  or  any 
other  ratio  of  the  velocity,  the  difference  will  be  in  the  same  ratio  of  the. 
whole  arc  :  and  the  contrary. 

COR.  4.  If  the  resistance  be  partly  in  the  simple  ratio  of  the  velocity, 
and  partly  in  the  duplicate  ratio  of  the  same,  the  difference  will  be  partly 
in  the  ratio  of  the  whole  arc,  and  partly  in  the  duplicate  ratio  of  it:  and 
the  contrary.  So  that  the  law  arid  ratio  of  the  resistance  will  be  the 
same  for  the  velocity  as  the  law  and  ratio  of  that  difference  for  the  length 
of  the  arc. 

COR.  5.  And  therefore  if  a  pendulum  describe  successively  unequal  arcs, 
and  we  can  find  the  ratio  of  the  increment  or  decrement  of  this  difference 
for  the  length  of  the  arc  described,  there  will  be  had  also  the  ratio  of  the 
increment  or  decrement  of  the  resistance  for  a  greater  or  less  velocity. 

GENERAL  SCHOLIUM. 

From  these  propositions  we  may  find  the  resistance  of  mediums  by  pen 
dulums  oscillating  therein.  I  found  the  resistance  of  the  air  by  the  fol 
lowing  experiments.  I  suspended  a  wooden  globe  or  ball  weighing  oT^ 
ounces  troy,  its  diameter  CJ  London  inches,  by  a  fine  thread  on  a  firm 
hook,  so  that  the  distance  between  the  hook  and  the  centre  of  oscillation  of 
the  globe  was  10|  feet.  I  marked  on  the  thread  a  point  10  feet  and  1  inch 
distant  from  the  centre  of  suspension  •  and  even  with  that  point  I  placed  a 
ruler  divided  into  inches,  by  the  help  whereof  I  observed  the  lengths  of  the 
arcs  described  by  the  pendulum.  Then  I  numbered  the  oscillations  ia 
which  the  globe  would  lose  -{-  part  of  its  motion.  If  the  pendulum  was 
drawn  aside  from  the  perpendicular  to  the  distance  of  2  inches,  and  thence 
let  go,  so  that  in  its  whole  descent  it  described  an  arc  of  2  inches,  and  in 
the  first  whole  oscillation,  compounded  of  the  descent  and  subsequent 
ascent,  an  arc  of  almost  4  inches,  the  same  in  164  oscillations  lost  j  part 
of  its  motion,  so  as  in  its  last  ascent  to  describe  an  arc  of  If  inches.  If 
in  the  first  descent  it  described  an  arc  of  4  inches,  it  lost  j  part  of  its  mo 
tion  in  121  oscillations,  so  as  in  its  last  ascent  to  describe  an  arc  of  3^ 
inches.  If  in  the  first  descent  it  described  an  arc  of  8, 16,  32,  or  64  inches, 
it  lost  |  part  of  its  motion  in  69,  35|,  18|-7  9|  oscillations,  respectively. 
Therefore  the  difference  between  the  arcs  described  in  the  first  descent  and 
the  last  ascent  was  in  the  1st,  2d,  3d,  4th,  5th,  6th  cases,  },  1.  1,  2,  4,  8 
inches  respectively.  Divide  those  differences  by  the  number  of  oscillations 
in  each  case,  and  in  one  mean  oscillation,  wherein  an  arc  of  3£ ,  7-|,  15,  30 


314  THE    MATHEMATICAL    PRINCIPLES  [BOOK    Jl. 

60,  120  inches  was  described,  the  difference  of  the  arcs  described  in  the 
descent  and  subsequent  ascent  will  be  ¥|^,  ^{^  e\>  T4r;  -sji  fir  parts  of  an 
inch,  respectively.  But  these  differences  in  the  greater  oscillations  are  in 
the  duplicate  ratio  of  the  arcs  described  nearly,  but  in  lesser  oscillations 
something  greater  than  in  that  ratio  ;  and  therefore  (by  Cor.  2,  Prop.  XXXI 
of  this  Book)  the  resistance  of  the  globe,  when  it  moves  very  swift,  is  in 
the  duplicate  ratio  of  the  velocity,  nearly;  and  when  it  moves  slowly, 
somewhat  greater  than  in  that  ratio. 

Now  let  V  represent  the  greatest  velocity  in  any  oscillation,  and  let  A, 
B,  and  C  be  given  quantities,  and  let  us  suppose  the  difference   of  the  arcs 

3^ 

to  be  AV  +  BV2  +  CV2.  Since  the  greatest  velocities  are  in  the  cycloid 
as  ^  the  arcs  described  in  oscillating,  and  in  the  circle  as  |  the  chords  of 
those  arcs ;  and  therefore  in  equal  arcs  are  greater  in  the  cycloid  than  in 
the  circle  in  the  ratio  of  |  the  arcs  to  their  chords ;  but  the  times  in  the 
circle  are  greater  than  in  the  cycloid,  in  a  reciprocal  ratio  of  the  velocity ; 
it  is  plain  that  the  differences  of  the  arcs  (which  are  as  the  resistance  and 
the  square  of  the  time  conjunctly)  are  nearly  the  same  in  both  curves :  for 
in  the  cycloid  those  differences  must  be  on  the  one  hand  augmented,  with 
the  resistance,  in  about  the  duplicate  ratio  of  the  arc  to  the  chord,  because 
of  the  velocity  augmented  in  the  simple  ratio  of  the  same ;  and  on  the 
other  hand  diminished,  with  the  square  of  the  time,  in  the  same  duplicate 
ratio.  Therefore  to  reduce  these  observations  to  the  cycloid,  we  must  take 
the  same  differences  of  the  arcs  as  were  observed  in  the  circle,  and  suppose 
the  greatest  velocities  analogous  to  the  half,  or  the  whole  arcs,  that  is,  to 
the  numbers  ±,  1,  2,  4,  8,  16.  Therefore  in  the  2d,  4th,  and  6th  cases,  put 
1,4,  and  1 6  for  V ;  and  the  difference  of  the  arcs  in  the  2d  case  will  become 

i  2 

* =  A  +  B  +  C;  in  the4th  case,  ^-  =  4A  +  SB  +  160  ;  in  the  6th 

121  OOj 

case,  ^  =  16A  +  64B  -f-  256C.     These   equations   reduced   give  A  = 

9? 
0,000091 6,  B  =-.  0,0010847,  and  C  =  0,0029558.     Therefore  the  difference 

of  the  arcs  is  as  0,0000916V  -f  0,0010847V*  +  0,0029558 V* :  and  there 
fore  since  (by  Cor.  Prop.  XXX,  applied  to  this  case)  the  re.-ist;mcc  of  the 
globe  in  the  middle  of  the  arc  described  in  oscillating,  where  the  velocity 

is  V,  is  to  its  weight  as  T7TAV  -f-  T\BV^  +  f  CV2  to  the  length  of  the 
pendulum,  if  for  A,  B,  and  C  you  put  the  numbers  found,  the  resistance  of 

the  globe  will  be  to  its  weight  as  0,0000583V  +  0,0007593V*  +  0,OJ22169V2 
to  the  length  of  the  pendulum  between  the  centre  of  suspension  and  the 
ruler,  that  is,  to  121  inches.  Therefore  since  V  in  the  second  case  repre 
sents  1,  in  the  4th  case  4,  and  in  the  6th  case  16,  the  resistance  will  be  to 
the  weight  of  the  globe,  in  the  2d  case,  as  0,0030345  to  121 ;  in  the  4th,  as 
0,041748  to  121 ;  in  the  6th,  as  0,61705  to  121. 


SEC.     VI.]  OF    NATURAL    PHILOSOPHY.  315 

The  arc,  which  the  point  marked  in  the  thread  described  in  the  6th  case, 

was  of  120  —  Q^,  or  119/g  inches.     And  therefore  since  the  radius  was 

ya 

121  inches,  and  the  length  of  the  pendulum  between  the  point  of  suspen 
sion  and  the  centre  of  the  globe  was  126  inches,  the  arc  which  the  centre  of 
the  globe  described  was  124/T  inches.  Because  the  greatest  velocity  of  the 
oscillating  body,  by  reason  of  the  resistance  of  the  air,  does  not  fall  on  the 
lowest  point  of  the  arc  described,  but  near  the  middle  place  of  the  whole 
arc,  this  velocity  will  be  nearly  the  same  as  if  the  globe  in  its  whole  descent 
in  a  non-resisting  medium  should  describe  62 ^  inches,  the  half  of  that  arc, 
and  that  in  a  cycloid,  to  which  we  have  above  reduced  the  motion  of  the 
pendulum;  and  therefore  that  velocity  will  be  equal  to  that  which  the 
globe  would  acquire  by  falling  perpendicularly  from  a  height  equal  to  the 
versed  sine  of  that  arc.  But  that  versed  sine  in  the  cycloid  is  to  that  arc 
62/2  as  the  same  arc  to  twice  the  length  of  the  pendulum  252,  and  there 
fore  equal  to  15,278  inches.  Therefore  the  velocity  of  the  pendulum  is  the 
same  which  a  body  would  acquire  by  falling,  and  in  its  fall  describing  a 
space  of  15,278  inches.  Therefore  with  such  a  velocity  the  globe  meets 
with  a  resistance  which  is  to  its  weight  as  0,61705  to  121,  or  (if  we  take 
that  part  only  of  the  resistance  which  is  in  the  duplicate  ratio  of  the  ve- 
loc.ty)  as  0,56752  to  121. 

I  found,  by  an  hydrostatical  experiment,  that  the  weight  of  this  wooden 
globe  was  to  the  weight  of  a  globe  of  water  of  the  same  magnitude  as  55 
to  97:  and  therefore  since  121  is  to  213,4  in  the  same  ratio,  the  resistance 
made  to  this  globe  of  water,  moving  forwards  with  the  above-mentioned 
velocity,  will  be  to  its  weight  as  0,56752  to  213,4,  that  is,  as  1  to  376^. 
Whence  since  the  weight  of  a  globe  of  water,  in  the  time  in  which  the 
globe  with  a  velocity  uniformly  continued  describes  a  length  of  30,556 
inches,  will  generate  all  that  velocity  in  the  falling  globe,  it  is  manifest 
that  the  force  of  resistance  uniformly  continued  in  the  same  time  will  take 
away  a  velocity,  which  will  be  less  than  the  other  in  the  ratio  of  1  to  376^-0, 

that  is,  the  rr^-r  part  of  the  whole  velocity.     And  therefore  in  the  time 

37  VSG 

Jiat  the  globe,  with  the  same  velocity  uniformly  continued,  would  describe 
the  length  of  its  semi-diameter,  or  3T\  inches,  it  would  lose  the  3^42  part 
of  its  motion. 

I  also  counted  the  oscillations  in  which  the  pendulum  lost  j  part  of  its 
motion.  In  the  following  table  the  upper  numbers  denote  the  length  of  the 
arc  described  in  the  first  descent,  expressed  in  inches  and  parts  of  an  inch ; 
the  middle  numbers  denote  the  length  of  the  arc  described  in  the  last  as 
cent  ;  and  in  the  lowest  place  are  the  numbers  of  the  oscillations.  I  give 
un  account  of  this  experiment,  as  being  more  accurate  than  that  in  which 


316  THE    MATHEMATICAL    PRINCIPLES  [BOOK    ll 

only  1  part  of  the  motion  was  lost.     I  leave  the  calculation  to  such  as  are 
disposed  to  make  it. 

First  descent   ...       2  4  8  16  32  64 

Last  ascent      .     .    ,       1|  3  6  12  24  48 

Numb.ofoscilL   .     .374         272         162i          83J  41f  22| 

I  afterward  suspended  a  leaden  globe  of  2  inches  in  diameter,  weighing 
26 1  ounces  troy  by  the  same  thread,  so  that  between  the  centre  of  the 
globe  and  the  point  of  suspension  there  was  an  interval  of  10^  feet,  and  1 
counted  the  oscillations  in  which  a  given  part  of  the  motion  was  lost.  The 
iirst  of  the  following  tables  exhibits  the  number  of  oscillations  in  which  J- 
part  of  the  whole  motion  was  lost ;  the  second  the  number  of  oscillations 
in  which  there  was  lost  \  part  of  the  same. 

First  descent  ....       1  2  4  8  16       32  64 

Last  ascent    ....         f  '  J  3^  7  14      28  56 

Numb,  of  oscilL       .     .  226         228         193         140  90^     53  30 

First  descent  ....       1  2  4  8  16       32  64 

Last  ascent     ....         £  1^  3  6  12       24  4S 

Nunib.  of  oscill.       .     .510         518^       420         318  204     12170 

Selecting  in  the  first  table  the  3d,  5th,  and  7th  observations,  and  express 
ing  the  greatest  velocities  in  these  observations  particularly  by  the  num 
bers  1,  4,  16  respectively, 'and  generally  by  the  quantity  V  as  above,  there 

will  come  out  in  ihe  3d  observation  ~-  =  A  +  B  +  C,  in  the  5th  obser- 


2  8 

vation  ^—  =  4A  4-  8B  +  16C.  in  the  7th  observation  ^--  ==  16A  4-  64B  t- 
,t(j  j  oU 

256C.  These  equations  reduced  give  A  =  0,001414,  B  ==  0,000297,  C  — 
0,000879.  And  thence  the  resistance  of  the  globe  moving  with  the  velocity 
V  will  be  to  its  weight  26^  ounces  in  the  same  ratio  as  0,0009V  + 

0,000208V*  +  0,000659V2  to  121  inches,  the  length  of  the  pendulum. 
And  if  we  regard  that  part  only  of  the  resistance  which  is  in  the  dupli 
cate  ratio  of  the  velocity,  it  will  be  to  the  weight  of  the  globe  as  0,000659V2 
to  121  inches.  But  this  part  of  the  resistance  in  the  first  experiment  was 
to  the  weight  oi'  the  wooden  globe  of  572-72  ounces  as  0,002217V2  to  121  ; 
and  thence  the  resistance  of  the  wooden  globe  is  to  the  resistance  of  the 
leaden  one  (their  velocities  being  equal)  as  57/2-  into  0,002217  to  26 J- 
into  0,000659,  that  is,  as  7|-  to  1.  The  diameters  of  the  two  globes  were 
6f  and  2  inches,  and  the  squares  of  these  are  to  each  other  as  47  £  and  4, 
or  11-J-f  and  1,  nearly.  Therefore  the  resistances  of  these  equally  swift 
globes  were  in  less  than  a  duplicate  ratio  of  the  diameters.  But  we  have 
not  yet  considered  the  resistance  of  the  thread,  which  was  certainly  very 
considerable,  and  ought  to  be  subducted  from  the  resistance  of  the  pendu 
lums  here  found.  I  could  not  determine  this  accurately,  but  I  found  il 


SEC.    VI.J  OF    NATURAL    PHILOSOPHY.  3 1/ 

greater  than  a  third  part  of  the  whole  resistance  of  the  lesser  pendulum  ; 
and  thence  I  gathered  that  the  resistances  of  the  globes,  when  the  resist 
ance  of  the  thread  is  subducted,  are  nearly  in  the  duplicate  ratio  of  their 
diameters.  For  the  ratio  of  7}  —  }  to  1  —  £,  or  l(H  to  1  is  not  very 
different  from  the  duplicate  ratio  of  the  diameters  1  L}f  to  I. 

Since  the  resistance  of  the  thread  is  of  less  moment  in  greater  globes,  I 
tried  the  experiment  also  with  a  globe  whose  diameter  was  ISf  inches. 
The  length  of  the  pendulum  between  the  point  of  suspension  and  the  cen 
tre  of  oscillation  was  122|  inches,  and  between  the  point  of  suspension  and 
the  knot  in  the  thread  109|  inches.  The  arc  described  by  the  knot  at  the 
first  descent  of  the  pendulum  was  32  inches.  The  arc  described  by  the 
same  knot  in  the  last  ascent  after  five  oscillations  was  2S  inches.  The 
sum  of  the  arcs,  or  the  whole  arc  described  in  one  mean  oscillation,  was  60 
inches.  The  difference  of  the  arcs  4  inches.  The  y1,,-  part  of  this,  or  the 
difference  between  the  descent  and  ascent  in  one  mean  oscillation,  is  f  of 
an  inch.  Then  as  the  radius  10(J|  to  the  radius  122^,  so  is  the  whole  arc 
of  60  inches  described  by  the  knot  in  one  mean  oscillation  to  the  whole  arc 
of  67}  inches  described  by  the  centre  of  the  globe  in  one  mean  oscillation ; 
and  so  is  the  difference  |  to  a  new  difference  0,4475.  If  the  length  of  the 
arc  described  were  to  remain,  and  the  length  of  the  pendulum  should  be 
augmented  in  the  ratio  of  126  to  122},  the  time  of  the  oscillation  would 
be  augmented,  and  the  velocity  of  the  pendulum  would  be  diminished  in 
the  subduplicate  of  that  ratio  ;  so  that  the  difference  0,4475  of  the  arcs  de 
scribed  in  the  descent  and  subsequent  ascent  would  remain.  Then  if  the 
arc  described  be  augmented  in  the  ratio  of  124  33T  to  67},  that  difference 
0.4475  would  be  augmented  in  the  duplicate  of  that  ratio,  and  so  would 
become  1,5295.  These  things  would  be  so  upon  the  supposition  that  the 
resistance  of  the  pendulum  were  in  the  duplicate  ratio  of  the  velocity. 
Therefore  if  the  pendulum  describe  the  whole  arc  of  12433T  inches,  and  its 
length  between  the  point  of  suspension  and  the  centre  of  oscillation  be  126 
inches,  the  difference  of  the  arcs  described  in  the  descent  and  subsequent 
ascent  would  be  1,5295  inches.  And  this  difference  multiplied  into  the 
weight  of  the  pendulous  globe,  which  was  208  ounces,  produces  318,136. 
Again ;  in  the  pendulum  above-mentioned,  made  of  a  wooden  globe,  when 
its  centre  of  oscillation,  being  126  inches  from  the  point  of  suspension,  de 
scribed  the  whole  arc  of  124 /T  inches,  the  difference  of  the  arcs  described 

in  the  descent  and  ascent  was  ^^   into  ^.      This   multiplied    into    the 

i/wi         y^ 

weight  of  the  globe,  which  was  57-272  ounces,  produces  49,396.  But  I  mul 
tiply  these  differences  into  the  weights  of  the  globes,  in  order  to  find  their 
resistances.  For  the  differences  arise  from  the  resistances,  and  are  as  the 
resistances  directly  and  the  weights  inversely.  Therefore  the  resistances 
are  as  the  numbers  318,136  and  49,396.  But  that  part  of  the  resistance 


31 S  THE    MATHEMATICAL    PRINCIPLES  [BOOK    1L 

of  the  lesser  globe,  which  is  in  the  duplicate  ratio  of  the  velocity,  was  to 
the  whole  resistance  as  0,56752  to-  0,61675,  that  is,  as  45,453  to  49,396 ; 
whereas  that  part  of  the  resistance  of  the  greater  globe  is  almost  equal  to 
its  whole  resistance ;  and  so  those  parts  are  nearly  as  318,136  and  45,453, 
that  is,  as  7  and  1.  But  the  diameters  of  the  globes  are  18f  and  6| ;  and 
their  squares  351T9¥  and  47 £J  are  as  7,438  and  1,  that  is,  as  the  resistances 
of  the  globes  7  and  1,  nearly.  The  difference  of  these  ratios  is  scarce 
greater  than  may  arise  from  the  resistance  of  the  thread.  Therefore  those 
parts  of  the  resistances  which  are,  when  the  globes  are  equal,  as  the  squares 
of  the  velocities,  are  also,  when  the  velocities  are  equal,  as  the  squares  of 
the  diameters  of  the  globes. 

But  the  greatest  of  the  globes  I  used  in  these  experiments  was  not  per 
fectly  spherical,  and  therefore  in  this  calculation  I  have,  for  brevity's  sake, 
neglected  some  little  niceties ;  being  not  very  solicitous  for  an  accurate 
calculus  in  an  experiment  that  was  not  very  accurate.  So  that  I  could 
wish  that  these  experiments  were  tried  again  with  other  globes,  of  a  larger 
size,  more  in  number,  and  more  accurately  formed ;  since  the  demonstra 
tion  of  a  vacuum  depends  thereon.  If  the  globes  be  taken  in  a  geometrical 
proportion,  as  suppose  whose  diameters  are  4,  8,  16,  32  inches;  one  may 
collect  from  the  progression  observed  in  the  experiments  what  would  hap 
pen  if  the  globes  were  still  larger. 

In  order  to  compare  the  resistances  of  different  fluids  with  each  other,  1 
made  the  following  trials.  I  procured  a  wooden  vessel  4  feet  long,  1  foot 
broad,  and  1  foot  high.  This  vessel,  being  uncovered,  1  filled  with  spring 
water,  and,  having  immersed  pendulums  therein,  I  made  them  oscillate  in 
the  water.  And  I  found  that  a  leaden  globe  weighing  166|  ounces,  and  in 
diameter  3f  inches,  moved  therein  as  it  is  set  down  in  the  following  table ; 
the  length  of  the  pendulum  from  the  point  of  suspension  to  a  certain 
point  marked  in  the  thread  being  126  inches,  and  to  the  centre  of  oscilla 
tion  134  f  inches. 
The  arc  described  in  } 

the  first  descent,  by 

a  point  marked  in  \  64    .    32    .    16    .   $    .    4    .    2    .    1    .    £    .    J 

the      thread      was  \ 

inches. 
The  arc  described  in  ) 

the  last  ascent  was  V  48   .   24   .    12   .   6    .   3   .    1|    .    £    .    f    .    T\ 

inches.  \ 

The  difference  of  the 

arcs,     proportional 

to  the  'motion  lost, 

was  inches. 
The  number  of  the  os 
cillations  in  water. 
The  number  of  the  os 
cillations  in  air. 


16 


«.  li   .   3  .  7  .  lH.12f.13j 


85i  .  287 . 535 


SEC.    VI.]  OF    NATURAL    PHILOSOPHY.  319 

In  the  experiments  of  the  4th  column  there  were  equal  motions  lost  in 
535  oscillations  made  in  the  air,  and  If  in  water.  The  oscillations  in  the 
air  were  indeed  a  little  swifter  than  those  in  the  water.  But  if  the  oscil 
lations  in  the  water  were  accelerated  in  such  a  ratio  that  the  motions  of 
the  pendulums  might  be  equally  swift  in  both  mediums,  there  would  be 
still  the  same  number  1  j  of  oscillations  in  the  water,  and  by  these  the 
same  quantity  of  motion  would  be  lost  as  before  ;  because  the  resistance  i> 
increased,  and  the  square  of  the  time  diminished  in  the  same  duplicate  ra 
tio.  The  pendulums,  therefore,  being  of  equal  velocities,  there  were  equal 
motions  lost  in  535  oscillations  in  the  air,  and  1}  in  the  water;  and  there 
fore  the  resistance  of  the  pendulum  in  the  water  is  to  its  resistance  in  the 
air  as  535  to  1  }.  This  is  the  proportion  of  the  whole  resistances  in  the 
case  of  the  4th  column. 

Now  let  AV  +  CV2  represent  the  difference  of  the  arcs  described  in  the 
descent  and  subsequent  ascent  by  the  globe  moving  in  air  with  the  greatest 
velocity  V  ;  and  since  the  greatest  velocity  is  in  the  case  of  the  4th  column 
to  the  greatest  velocity  in  the  case  of  the  1st  column  as  1  to  8  ;  and  that 
difference  of  the  arcs  in  the  case  of  the  4th  column  to  the  difference  in  the 

2         16 

case  of  the  1st  column  as  ^      to      7,  or  as  86  J  to  4280  ;  put   in    these 


cases  1  and  8  for  the  velocities,  and  85  1  and  4280  for  the  differences  of 
the  arcs,  and  A  +  C  will  be  —  S5|,  and  8A  -f  640  ==•  4280  or  A  +  SC 
=  535  ;  and  then  by  reducing  these  equations,  there  will  come  out  TC  = 
449^  and  C  =  64T\  and  A  =  21f  ;  and  therefore  the  resistance,  which  is 
as  TVAV  +  fCV2,  will  become  as  13T6TV  +  48/^Y2.  Therefore  in  the 
case  of  the  4th  column,  where  the  velocity  was  1,  the  whole  resistance  is  to 
its  part  proportional  to  the  square  of  the  velocity  as  13T6T  +  48/F  or 
61  }f  to  48/e  ;  and  therefore  the  resistance  of  the  pendulum  in  water  is  to 
that  part  of  the  resistance  in  air,  which  is  proportional  to  the  square  of  the 
velocity,  and  which  in  swift  motions  is  the  only  part  that  deserves  consid 
eration,  as  61}^  to  4S/g  and  535  to  1}  conjunctly,  that  is,  as  571  to  1. 
If  the  whole  thread  of  the  pendulum  oscillating  in  the  water  had  been  im 
mersed,  its  resistance  would  have  been  still  greater  ;  so  that  the  resistance 
of  the  pendulum  oscillating  in  the  water,  that  is,  that  part  which  is  pro 
portional  to  the  square  of  the  velocity,  and  which  only  needs  to  be  consid 
ered  in  swift  bodies,  is  to  the  resistance  of  the  same  whole  pendulum,  oscil 
lating  in  air  with  the  same  velocity,  as  about  850  to  1,  that  is  as,  the  den 
sity  of  water  to  the  density  of  air,  nearly. 

In  this  calculation  we  ought  also  to  have  taken  in  that  part  of  the  re 
sistance  of  the  pendulum  in  the  water  which  was  as  the  square  of  the  ve 
locity  ;  but  I  found  (which  will  perhaps  seem  strange)  that  the  resistance 
in  the  water  was  augmented  in  more  than  a  duplicate  ratio  of  the  velocity. 
In  searching  after  the  cause,  I  thought  upon  this,  that  the  vessel  was  toe 


320  THE    MATHEMATICAL    PRINCIPLES  [BOOK    II. 

narrow  for  the  magnitude  of  the  pendulous  globe,  and  by  its  narrowness 
obstructed  the  motion  of  the  water  as  it  yielded  to  the  oscillating  globe. 
For  when  I  immersed  a  pendulous  globe,  whose  diameter  was  one  inch  only, 
the  resistance  was  augmented  nearly  in  a  duplicate  ratio  of  the  velocity, 
I  tried  this  by  making  a  pendulum  of  two  globes,  of  which  the  lesser  and 
lower  oscillated  in  the  water,  and  the  greater  and  higher  was  fastened  to 
the  thread  just  above  the  water,  and,  by  oscillating  in  the  air,  assisted  the 
motion  of  the  pendulum,  and  continued  it  longer.  The  experiments  made 
by  this  contrivance  proved  according  to  the  following  table. 
Arc  descr.  in  first  descent  .  .16.8.  4.  2.1.1.1 
Arc  descr.  in  last  ascent  .  .  12  .  6  .  3  .  li  .  J  .  |  .  T3F 
Dif.  of  arcs,  proport.  to  1  .  pi  i 

motion  lost  $  T     •     «r     •  T* 

Number    of    oscillations...     3f  .    6j  .    12^.  211  .     34  .  53  .  62) 

In  comparing  the  resistances  of  the  mediums  with  each  other,  I  also 
caused  iron  pendulums  to  oscillate  in  quicksilver.  The  length  of  the  iron 
wire  was  about  3  feet,  and  the  diameter  of  the  pendulous  globe  about  i  of 
an  inch.  To  the  wire,  just  above  the  quicksilver,  there  was  fixed  another 
leaden  globe  of  a  bigness  sufficient  to  continue  the  motion  of  the  pendulum 
for  some  time.  Then  a  vessel,  that  would  hold  about  3  pounds  of  quick 
silver,  was  filled  by  turns  with  quicksilver  and  common  water,  that,  by 
making  the  pendulum  oscillate  successively  in  these  two  different  fluids,  I 
might  find  the  proportion  of  their  resistances ;  and  the  resistance  of  the 
quicksilver  proved  to  be  to  the  resistance  of  water  as  about  13  or  14  to  1  ; 
that  is.  as  the  density  of  quicksilver  to  the  density  of  water.  When  I  made 
use  of  a  pendulous  globe  something  bigger,  as  of  one  whose  diameter  was 
about  ^  or  |  of  an  inch,  the  resistance  of  the  quicksilver  proved  to  be  to 
the  resistance  of  the  water  as  about  12  or  10  to  1.  But  the  former  experi 
ment  is  more  to  be  relied  on,  because  in  the  latter  the  vessel  was  too  nar 
row  in  proportion  to  the  magnitude  of  the  immersed  globe;  for  the  vessel 
ought  to  have  been  enlarged  together  with  the  globe.  I  intended  to  have 
repeated  these  experiments  with  larger  vessels,  and  in  melted  metals,  and 
other  liquors  both  cold  and  hot ;  but  I  had  not  leisure  to  try  all:  and  be 
sides,  from  what  is  already  described,  it  appears  sufficiently  that  the  resist 
ance  of  bodies  moving  swiftly  is  nearly  proportional  to  the  densities  of 
the  fluids  in  which  they  move.  I  do  not  say  accurately ;  for  more  tena 
cious  fluids,  of  equal  density,  will  undoubtedly  resist  more  than  those  that 
are  more  liquid ;  as  cold  oil  more  than  warm,  warm  oil  more  than  rain 
water,  and  water  more  than  spirit  of  wine.  But  in  liquors,  which  are  sen 
sibly  fluid  enough,  as  in  air,  in  salt  and  fresh  water,  in  spirit  of  wine,  of 
turpentine,  and  salts,  in  oil  cleared  of  its  fseces  by  distillation  and  warmed, 
in  oil  of  vitriol,  and  in  mercury,  and  melted  metals,  and  any  other  such 
like,  that  are  fluid  enough  to  retaia  for  some  time  the  motion  impressed 


SEC.    VI.J  OF    NATURAL    PHILOSOPHY.  321 

upon  them  by  the  agitation  of  the  vessel,  and  which  being  poured  out  are 
easily  resolved  into  drops,  I  doubt  not  but  the  rule  already  laid  down  may 
be  accurate  enough,  especially  if  the  experiments  be  made  with  larger 
pendulous  bodies  and  more  swiftly  moved. 

Lastly,  since  it  is  the  opinion  of  some  that  there  is  a  certain  ^ethereal 
medium  extremely  rare  and  subtile,  which  freely  pervades  the  pores  of  all 
bodies ;  and  from  such  a  medium,  so  pervading  the  pores  of  bodies,  some  re 
sistance  must  needs  arise;  in  order  to  try  whether  the  resistance,  which  wre 
experience  in  bodies  in  motion,  be  made  upon  their  outward  superficies  only, 
or  whether  their  internal  parts  meet  with  any  considerable  resistance  upon 
their  superficies,  I  thought  of  the  following  experiment  I  suspended  a 
round  deal  box  by  a  thread  11  feet  long,  on  a  steel  hook,  by  means  of  a  ring 
of  the  s-ime  metal,  so  as  to  make  a  pendulum  of  the  aforesaid  length.  The 
hook  had  a  sharp  hollowr  edge  on  its  upper  part,  so  that  the  upper  arc  of 
the  ring  pressing  on  the  edge  might  move  the  more  freely ;  and  the  thread 
was  fastened  to  the  lower  arc  of  the  ring.  The  pendulum  being  thus  pre 
pared,  I  drew  it  aside  from  the  perpendicular  to  the  distance  of  about  6 
feet,  and  that  in  a  plane  perpendicular  to  the  edge  of  the  hook,  lest  the 
ring,  while  the  pendulum  oscillated,  should  slide  to  and  fro  on  the  edge  of 
the  hook :  for  the  point  of  suspension,  in  which  the  ring  touches  the  hook, 
ought  to  remain  immovable.  I  therefore  accurately  noted  the  place  to 
which  the  pendulum  was  brought,  and  letting  it  go,  I  marked  three  other 
places,  to  which  it  returned  at  the  end  of  the  1st,  2d,  and  3d  oscillation. 
This  I  often  repeated,  that  I  might  find  those  places  as  accurately  as  pos 
sible.  Then  I  filled  the  box  with  lead  and  other  heavy  metals  that  were 
near  at  hand.  But,  first,  I  weighed  the  box  when  empty,  and  that  pnrt  of 
the  thread  that  went  round  it,  and  half  the  remaining  part,  extended  be 
tween  the  hook  and  the  suspended  box ;  for  the  thread  so  extended  always 
acts  upon  the  pendulum,  when  drawn  aside  from  the  perpendicular,  with  half 
its  weight.  To  this  weight  I  added  the  weight  of  the  air  contained  in  the 
box  And  this  whole  weight  was  about  -fj  of  the  weight  of  the  box  when 
filled  writh  the  metals.  Then  because  the  box  when  full  of  the  metals,  by  ex 
tending  the  thread  with  its  weight,  increased  the  length  of  the  pendulum, 
f  shortened  the  thread  so  as  to  make  the  length  of  the  pendulum,  when  os 
cillating,  the  same  as  before.  Then  drawing  aside  the  pendulum  to  the 
place  first  marked,  and  letting  it  go,  I  reckoned  about  77  oscillations  before 
the  box  returned  to  the  second  mark,  and  as  many  afterwards  before  it  came 
to  the  third  mark,  and  as  many  after  that  before  it  came  to  the  fourth 
xnark.  From  whence  I  conclude  that  the  whole  resistance  of  the  box,  when 
full,  had  not  a  greater  proportion  to  the  resistance  of  the  box,  when  empty, 
than  78  to  77.  For  if  their  resistances  were  equal,  the  box,  when  full,  by 
reason  of  its  vis  insita,  which  was  78  times  greater  than  the  vis  tfuritoof 
the  same  when  empty,  ought  to  have  continued  its  oscillating  motion  so 

21 


322  THE    MATHEMATICAL    PRINCIPLES  |  BOOK    II. 

much  the  longer,  and  therefore  to  have  returned  to  those  marks  at  the  end 
of  78  oscillations.  But  it  returned  to  them  at  the  end  of  77  oscillations. 

Let,  therefore,  A  represent  the  resistance  of  the  box  upon  its  external 
superficies,  and  B  the  resistance  of  the  empty  box  on  its  internal  superficies ; 
and  if  the  resistances  to  the  internal  parts  of  bodies  equally  swift  be  as  the 
matter,  or  the  number  of  particles  that  are  resisted,  then  78B  will  be  the 
resistance  made  to  the  internal  parts  of  the  box,  when  full ;  and  therefore 
the  whole  resistance  A  +  B  of  the  empty  box  will  be  to  the  whole  resist 
ance  A  +  7SB  of  the  full  box  as  77  to  78,  and,  by  division,  A  +  B  to  77B 
as  77  to  1 ;  and  thence  A  +  B  to  B  as  77  X  77  to  1,  and,  by  division 
again,  A  to  B  as  5928  to  1.  Therefore  the  resistance  of  the  empty  box  in 
its  internal  parts  will  be  above  5000  times  less  than  the  resistance  on  its 
external  superficies.  This  reasoning  depends  upon  the  supposition  that  the 
greater  resistance  of  the  full  box  arises  not  from  any  other  latent  cause, 
but  only  from  the  action  of  some  subtile  fluid  upon  the  included  metal. 

This  experiment  is  related  by  memory,  the  paper  being  lost  in  which  I 
had  described  it ;  so  that  I  have  been  obliged  to  omit  some  fractional  parts, 
which  are  slipt  out  of  my  memory ;  and  I  have  no  leisure  to  try  it  again. 
The  first  time  I  made  it,  the  hook  being  weak,  the  full  box  was  retarded 
sooner.  The  cause  I  found  to  be,  that  the  hook  was  not  strong  enough  to 
bear  the  weight  of  the  box :  so  that,  as  it  oscillated  to  and  fro,  the  hook 
was  bent  sometimes  this  and  sometimes  that  way.  I  therefore  procured  a 
hook  of  sufficient  strength,  so  that  the  point  of  suspension  might  remain 
unmoved,  and  then  all  things  happened  as  is  above  described. 


SEC.    VI  I.]  OF    NATURAL    PHILOSOPHY.  323 

SECTION  VII. 

Of  the,  motion  of  fluids,  and  the  resistance  made  to  projected  bodies. 

PROPOSITION  XXXII.     THEOREM  XXVI. 

Suppose  two  similar  systems  of  bodies  consisting  of  an  equal  number  of 
particles,  and  let  the  correspondent  particles  be  similar  and  propor 
tional,  each  in,  one  system  to  each  in  the  other,  and  have  a  like  situa 
tion  among  themselves,  and  the  same  given  ratio  of  density  to  each 
other ;  and  let  them  begin  to  move  anwng  themselves  in  proportional 
times,  and  with  like  motions  (that  is,  those  in  one  system  among  one 
another,  and  those  in  the  other  among  one  another).  And  if  the  par 
ticles  that  are  in  the  same  system  do  not  touch  otte  another,  except  ir 
the  moments  of  reflexion  ;  nor  attract,  nor  repel  each  other,  except  with 
accelerativeforc.es  that  are  as  the  diameters  of  the  correspondent  parti 
cles  inversely,  and  the  squares  of  the  velocities  directly  ;  I  say,  that  the 
particles  of  those  systems  will  continue  to  move  among  themselves  wit  It 
like  motions  and  in  proportional  times. 

Like  bodies  in  like  situations  are  said  to  be  moved  among  themselves 
with  like  motions  and  in  proportional  times,  when  their  situations  at  the 
end  of  those  times  are  always  found  alike  in  respect  of  each  other ;  as  sup 
pose  we  compare  the  particles  in  one  system  with  the  correspondent  parti 
cles  in  the  other.  Hence  the  times  will  be  proportional,  in  which  similar 
and  proportional  parts  of  similar  figures  will  be  described  by  correspondent 
particles.  Therefore  if  we  suppose  two  systems  of  this  kind;  the  corre 
spondent  particles,  by  reason  of  the  similitude  of  the  motions  at  their 
beginning,  will  continue  to  be  moved  with  like  motions,  so  long  as  they 
move  without  meeting  one  another ;  for  if  they  are  acted  on  by  no  forces, 
they  will  go  on  uniformly  in  right  lines,  by  the  1st  Law.  But  if  they  do 
agitate  one  another  with  some  certain  forces,  and  those  forces  are  as  the 
diameters  of  the  correspondent  particles  inversely  and  the  squares  of  the 
velocities  directly,  then,  because  the  particles  are  in  like  situations,  and 
their  forces  are  proportional,  the  whole  forces  with  which  correspondent 
particles  are  agitated,  and  which  are  compounded  of  each  of  the  agitating 
forces  (by  Corol.  2  of  the  Laws),  will  have  like  directions,  and  have  the 
same  effect  as  if  they  respected  centres  placed  alike  among  the  particles ; 
and  those  whole  forces  will  be  to  each  other  as  the  several  forces  which 
compose  them,  that  is,  as  the  diameters  of  the  correspondent  particles  in 
versely,  and  the  squares  of  the  velocities  directly  :  and  therefore  will  cans** 


3^4  THE    MATHEMATICAL    PRINCIPLES  [BOOK  11. 

correspondent  particles  to  continue  to  describe  like  figures.  These  things 
will  be  so  (by  Cor.  1  and  S,  Prop.  IV.;  Book  1),  if  those  centres  are  at  rest 
but  if  they  are  moved,  yet  by  reason  of  the  similitude  of  the  translations, 
their  situations  among  the  particles  of  the  system  will  remain  similar ,  so 
that  the  changes  introduced  into  the  figures  described  by  the  particles  will 
still  be  similar.  So  that  the  motions  of  correspondent  and  similar  par 
ticles  will  continue  similar  till  their  first  meeting  with  each  other ;  and 
thence  will  arise  similar  collisions,  and  similar  reflexions;  which  will  again 
beget  similar  motions  of  the  particles  among  themselves  (by  what  was  just 
now  shown),  till  they  mutually  fall  upon  one  another  again,  and  so  on  ad 
infinitum. 

COR.  1.  Hence  if  any  two  bodies,  which  are  similar  and  in  like  situations 
to  the  correspondent  particles  of  the  systems,  begin  to  move  amongst  them 
in  like  manner  and  in  proportional  times,  and  their  magnitudes  and  densi 
ties  be  to  each  other  as  the  magnitudes  and  densities  of  the  corresponding 
particles,  these  bodies  will  continue  to  be  moved  in  like  manner  and  in 
proportional  times:  for  the  case  of  the  greater  parts  of  both  systems  and  of 
the  particles  is  the  very  same. 

COR.  2.  And  if  all  the  similar  and  similarly  situated  parts  of  both  sys 
tems  be  at  rest  among  themselves ;  and  two  of  them,  which  are  greater  than 
the  rest,  and  mutually  correspondent  in  both  systems,  begin  to  move  in 
lines  alike  posited,  with  any  similar  motion  whatsoever,  they  will  excite 
similar  motions  in  the  rest  of  the  parts  of  the  systems,  and  will  continue 
to  move  among  those  parts  in  like  manner  and  in  proportional  times  ;  and 
will  therefore  describe  spaces  proportional  to  their  diameters. 


PROPOSITION  XXXIII.     THEOREM  XXVII. 

The  same  things  biting  supposed,  I  say,  that  the  greater  parts  of  the 
systems  are  resisted  in  a  ratio  compounded  of  the  duplicate  ratio  of 
their  velocities,  and  the  duplicate  ratio  of  their  diameters,  and  Ihe  sim 
ple  ratio  of  the  density  of  the  parts  of  the  systems. 
For  the  resistance  arises  partly  from  the  centripetal  or  centrifugal,  forces 
with  which  the  particles  of  the  system  mutually  act  on  each  other,  partly 
from  the  collisions  and  reflexions  of  the  particles  and   the  greater  parts. 
The  resistances  of  the  first  kind  are  to   each   other  as  the  whole  motive 
forces  from  which  they  arise,  that  is,  as  the  whole  accelerative  forces  and 
the  quantities  of  matter  in  corresponding  parts  ;    that  is   (by  the  sup 
position),  as  the  squares  of  the  velocities  directly,  and  the  distances  of  the 
corresponding  particles  inversely,  and  the  quantities  of  matter  in  the  cor 
respondent  parts  directly  :  and  therefore  since  the  distances  of  the  parti 
cles  in  one  system  are  to  the  correspondent  distances  of  the  particles  of  the 
;ther  S3  the  diameter  of  one  particle  or  part  in  *he  former  system  to  the 


SEC.    VII.]  OF    NATURAL    PHILOSOPHY.  C>2" 

diameter  of  the  correspondent  particle  or  part  in  the  other,  and  since  the 
quantities  of  matter  are  as  the  densities  of  the  parts  and  the  cubes  of  the 
diameters ;  the  resistances  arc  to  each  other  as  the  squares  of  the  velocities 
and  the  squares  of  the  diameters  and  the  densities  of  the  parts  of  the  sys 
tems.  Q.E.D.  The  resistances  of  the  latter  sort  are  as  the  number  of 
sorrespondent  reflexions  and  the  forces  of  those  reflexions  conjunctly ;  but 
the  number  of  the  reflexions  are  to  each  other  as  the  velocities  of  the  cor 
responding  parts  directly  and  the  spaces  between  their  reflexions  inversely. 
And  the  forces  of  the  reflexions  are  as  the  velocities  and  the  magnitudes 
and  the  densities  of  the  corresponding  parts  conjunctly ;  that  is,  as  the  ve 
locities  and  the  cubes  of  the  diameters  and  the  densities  of  the  parts.  And, 
joining  all  these  ratios,  the  resistances  of  the  corresponding  parts  are  to 
each  other  as  the  squares  of  the  velocities  and  the  squares  of  the  diameters 
and  the  densities  of  the  parts  conjunctly.  Q.E.T). 

COR.  1.  Therefore  if  those  systems  are  two  elastic  fluids,  like  our  air, 
and  their  parts  are  at  rest  among  themselves  ;  and  two  similar  bodies  pro 
portional  in  magnitude  and  density  to  the  parts  of  the  fluids,  and  similarly 
gituated  among  those  parts,  be  any  how  projected  in  the  direction  of  lines 
similarly  posited  ;  and  the  accelerative  forces  with  which  the  particles  of 
the  fluids  mutually  act  upon  each  other  are  as  the  diameters  of  the  bodies 
projected  inversely  and  the  squares  of  their  velocities  directly  ;  those  bodies 
will  excite  similar  motions  in  the  fluids  in  proportional  times,  and  will  de 
scribe  similar  spaces  and  proportional  to  their  diameters. 

COR.  2.  Therefore  in  the  same  fluid  a  projected  body  that  moves  swiftly 
meets  with  a  resistance  that  is,  in  the  duplicate  ratio  of  its  velocity,  nearly. 
For  if  the  forces  with  which  distant  particles  act  mutually  upon  one 
another  should  be  augmented  in  the  duplicate  ratio  of  the  velocity,  the 
projected  body  would  be  resisted  in  the  same  duplicate  ratio  accurately ; 
and  therefore  in  a  medium,  whose  parts  when  at  a  distance  do  not  act  mu 
tually  with  any  force  on  one  another,  the  resistance  is  in  the  duplicate  ra 
tio  of  the  velocity  accurately.  Let  there  be,  therefore,  three  mediums  A, 
B,  C,  consisting  of  similar  and  equal  parts  regularly  disposed  at  equal 
distances.  Let  the  parts  of  the  mediums  A  and  B  recede  from  each  other 
with  forces  that  are  among  themselves  as  T  and  V ;  and  let  the  parts  of 
the  medium  C  be  entirely  destitute  of  any  such  forces.  And  if  four  equal 
bodies  D,  E,  P7  G,  move  in  these  mediums,  the  two  first  D  and  E  in  the 
two  first  A  and  B,  and  the  other  two  P  and  G  in  the  third  C ;  and  if  the 
velocity  of  the  body  D  be  to  the  velocity  of  the  body  E,  and  the  velocity 
of  the  body  P  to  the  velocity  of  the  body  G,  in  the  subduplicate  ratio  of 
the  force  T  to  the  force  V ;  the  resistance  of  the  body  D  to  the  resistance 
of  the  body  E,  and  the  resistance  of  the  body  P  to  the  resistance  of  the 
body  G,  will  be  in  the  duplicate  ratio  of  the  velocities  ;  and  therefore  the 
resistance  of  the  body  D  will  be  to  the  resistance  of  the  body  P  as  the  re- 


326  THE    MATHEMATICAL    PRINCIPLES  [BOOK    II 

sistance  of  the  body  E  to  the  resistance  of  the  body  G.  Let  the  bodies  1) 
and  F  be  equally  swift,  as  also  the  bodies  E  and  G ;  and,  augmenting  the 
velocities  of  the^bodies  D  arid  F  in  any  ratio,  and  diminishing  the  forces 
of  the  particles  of  the  medium  B  in  the  duplicate  of  the  same  ratio,  the 
medium  B  will  approach  to  the  form  and  condition  of  the  medium  C  at 
pleasure ;  and  therefore  the  resistances  of  the  equal  and  equally  swift 
bodies  E  and  G  in  these  mediums  will  perpetually  approach  to  equality 
so  that  their  difference  will  at  last  become  less  than  any  given.  There 
fore  since  the  resistances  of  the  bodies  D  and  F  are  to  each  other  as  the 
resistances  of  the  bodies  E  and  G,  those  will  also  in  like  manner  approach 
to  the  ratio  of  equality.  Therefore  the  bodies  1)  and  F,  when  they  move 
with  very  great  swiftness,  meet  with  resistances  very  nearly  equal;  and 
therefore  since  the  resistance  of  the  body  F  is  in  a  duplicate  ratio  of  the 
velocity,  the  resistance  of  the  body  D  will  be  nearly  in  the  same  ratio. 

Con.  3.  The  resistance  of  a  body  moving  very  swift  in  an  elastic  fluid 
is  almost  the  same  as  if  the  parts  of  the  fluid  were  destitute  of  their  cen 
trifugal  forces,  and  did  not  fly  from  each  other;  if  so  be  that  the  elasti 
city  of  the  fluid  arise  from  the  centrifugal  forces  of  the  particles,  and  the 
velocity  be  so  great  as  not  to  allow  the  particles  time  enough  to  act. 

COR.  4.  Therefore,  since  the  resistances  of  similar  and  equally  swift 
bodies,  in  a  medium  whose  distant  parts  do  not  fly  from  each  other,  are  as 
the  squares  of  the  diameters,  the  resistances  made  to  bodies  moving  with 
very  great  and  equal  velocities  in  an  elastic  fluid  will  be  as  the  squares  of 
the  diameters,  nearly. 

COR.  5.  And  since  similar,  equal,  and  equally  swift  bodies,  moving 
through  mediums  of  the  same  density,  whose  particles  do  not  fly  from  each 
other  mutually,  will  strike  against  an  equal  quantity  of  matter  in  equal 
times,  whether  the  particles  of  which  the  medium  consists  be  more  and 
smaller,  or  fewer  and  greater,  and  therefore  impress  on  that  matter  an  equal 
quantity  of  motion,  and  in  return  (by  the  3d  Law  of  Motion)  suffer  an 
equal  re-action  from  the  same,  that  is,  are  equally  resisted ;  it  is  manifest, 
also,  that  in  elastic  fluids  of  the  same  density,  when  the  bodies  move  with 
extreme  swiftness,  their  resistances  are  nearly  equal,  whether  the  fluids 
consist  of  gross  parts,  or  of  parts  ever  so  subtile.  For  the  resistance  of 
projectiles  moving  with  exceedingly  great  celerities  is  not  much  diminished 
by  the  subtilty  of  the  medium. 

COR.  G.  All  these  things  are  so  in  fluids  whose  elastic  force  takes  its  rise 
from  the  centrifugal  forces  of  the  particles.  But  if  that  force  arise  from 
some  other  cause,  as  from  the  expansion  of  the  particles  after  the  manner 
of  wool,  or  the  boughs  of  trees,  or  any  other  cause,  by  which  the  particles 
are  hindered  from  moving  freely  among  themselves,  the  resistance,  by 
reason  of  the  lesser  fluidity  of  the  medium,  will  be  greater  than  in  the 
Corollaries  above. 


SEC.  VII. 


OF    NATURAL    PHILOSOPHY. 


32? 


K 


L,     P 


O 


PROPOSITION  XXXIV.     THEOREM  XXV1I1. 

If  iu  a  rare  medium,  consisting  of  equal  particles  freely  disposed  at 
equal  distances  from  each  other,  a  globe  and  a  cylinder  described  on 
equal  diameters  move  with  equal  velocities  in  the.  direction  of  the  axis 
of  the  cylinder,  the  resistance  of  the  globe  ivill  be  but  half  so  great  an 
that  of  the  cylinder. 
For  since  the  action  of  the  medi 
um  upon  the  body  is  the  same  (by 
Cor.  5  of  the  Laws)  whether  the  body 
move  in  a  quiescent  medium,  or 
whether  the  particles  of  the  medium 
impinge  with  the  same  velocity  upon 
the  quiescent  body,  let  us  consider 
the  body  as  if  it  were  quiescent,  and 
see  with  what  force  it  would  be  im- 
pelled  by  the  moving  medium.  Let,  therefore,  ABKI  represent  a  spherical 
body  described  from  the  centre  C  with  the  semi-diameter  CA,  and  let  the 
particles  of  the  medium  impinge  with  a  given  velocity  upon  that  spherical 
body  in  the  directions  of  right  lines  parallel  to  AC :  and  let  FB  be  one  of 
those  right  lines.  In  FB  take  LB  equal  to  the  semi-diameter  CB,  and 
draw  BI)  touching  the  sphere  in  B.  Upon  KG  and  BD  let  fall  the  per 
pendiculars  BE,  LD ;  and  the  force  with  which  a  particle  of  the  medium, 
impinging  on  the  globe  obliquely  in  the  direction  FB,  would  strike  the 
globe  in  B,  will  be  to  the  force  with  which  the  same  particle,  meeting  the 
cylinder  ONGQ,  described  about  the  globe  with  the  axis  ACI,  would  strike 
it  perpendicularly  in  b,  as  LD  to  LB,  or  BE  to  BC.  Again ;  the  efficacy 
of  this  force  to  move  the  globe,  according  to  the  direction  of  its  incidence 
FB  or  AC,  is  to  the  efficacy  of  the  same  to  move  the  globe,  according  to 
the  direction  of  its  determination,  that  is,  in  the  direction  of  the  right  line 
BC  in  which  it  impels  the  globe  directly,  as  BE  to  BC.  And,  joining 
these  ratios,  the  efficacy  of  a  particle,  falling  upon  the  globe  obliquely  in 
the  direction  of  the  right  line  FBy  to  move  the  globe  in  the  direction  of  its 
incidence,  is  to  the  efficacy  of  the  same  particle  falling  in  the  same  line 
perpendicularly  on  the  cylinder,  to  move  it  in  the  same  direction,  as  BE2 
to  BC3.  Therefore  if  in  6E,  which  is  perpendicular  to  the  circular  base  of 
the  cylinder  NAO,  and  equal  to  the  radius  AC,  we  take  £H  equal  to 

BEa 

-— • ;  then  6H  will  be  to  6E  as  the  effect  of  the  particle  upon  the  globe  t<? 
\~i\j 

the  effect  of  the  particle  upon  the  cylinder.  Arid  therefore  the  solid  which 
is  formed  by  all  the  right  lines  6H  will  be  to  the  solid  formed  by  all  the 
right  lines  />E  as  the  effect  of  all  the  particles  upon  the  globe  to  the  effect 
of  all  the  particles  upon  the  cylinder.  But  the  former  of  these  solids  is  a 


328 


THE    MATHEiAlATICAL    PRINCIPLES 


[BooK  li. 


paraboloid  whose  vertex  is  C,  its  axis  CA,  and  latus  rectum  CA,  and  the 
latter  solid  is  a  cylinder  circumscribing  the  paraboloid ;  and  it  is  knowr 
that  a  paraboloid  is  half  its  circumscribed  cylinder.  Therefore  the  whole 
force  of  the  medium  upon  the  globe  is  half  of  the  entire  force  of  the  same 
upon  the  cylinder.  And  therefore  if  the  particles  of  the  medium  are  at 
rest,  and  the  cylinder  and  globe  move  with  equal  velocities,  the  resistance 
of  the  globe  will  be  half  the  resistance  of  the  cylinder.  Q.E.D. 

SCHOLIUM. 

By  the  same  method  other  figures  may  be  compared  together  as  to  their 
resistance;  and  those  may  be  found  which  are  most  apt  to  continue  their 
motions  in  resisting  mediums.  As  if  upon  the  circular  base  CEBH  from 
the  centre  O,  with  thy  radius  OC,  and  the  altitude  OD,  one  would  construct 
a  frustum  CBGF  of  a  cone,  which  should  meet  with  less  resistance  than 
any  other  frustum  constructed  with  the  same  base  and  altitude,  and  going 
forwards  towards  D  in  the  direction  of  its  axis :  bisect  the  altitude  OD  in 
U,,  and  produce  OQ,  to  S,  so  that  QS  may  be  equal  to  Q,C,  and  S  will  be 
the  vertex  of  the  cone  whose  frustum  is  sought. 
r 


J 


Whence,  by  the  bye,  since  the  angle  CSB  is  always  acute,  it  follows,  that, 
if  the  solid  ADBE  be  generated  by  the  convolution  of  an  elliptical  or  oval 
figure  ADBE  about  its  axis  AB,  and  the  generating  figure  be  touched  by- 
three  right  lines  FG,  GH,  HI,  in  the  points  F,  B,  and  I,  so  that  GH  shall 
be  perpendicular  to  the  axis  in  the  point  of  contact  B,  arid  FG,  HI  may  be 
inclined  to  GH  in  the  angles  FGB,  BHI  of  135  degrees:  the  solid  arising 
from  the  convolution  of  the  figure  ADFGH1E  about  the  same  axis  AB 
will  be  less  resisted  than  the  former  solid;  if  so  be  that  both  move  forward 
in  the  direction  of  their  axis  AB,  and  that  the  extremity  B  of  each  go 
foremost.  Which  Proposition  I  conceive  may  be  of  use  in  the  building  of 
ships. 

If  the  figure  DNFG  be  such  a  curve,  that  if,  from  any  point  thereof,  as 
N,  the  perpendicular  NM  be  let  fall  on  the  axis  AB,  and  from  the  given 
point  G  there  be  drawn  the  right  line  GR  parallel  to  a  right  line  touching 
the  figure  in  N,  and  cutting  the  axis  produced  in  R,  MN  becomes  to  GR 
as  GR,3  to  4BR  X  GB2,  the  solid  described,  by  the  revolution  of  this  figure 


SEC.    Vll.J  OF    NATURAL    PHILOSOPHY.  32S 

about  its  axis  AB,  moving  in  the  before-mentioned  rare  medium  from  A 
towards  B,  will  be  less  resisted  than  any  other  circular  solid  whatsoever, 
described  of  the  same  length  and  breadth. 

PROPOSITION  XXXV.     PROBLEM  VII. 

If  a  rare  medium  consist  of  very  small  quiescent  particles  of  equal  mag 
nitudes,  and  freely  disposed  at  equal  distances  from  one  another :  to 
jind  the  resistance  of  a  globe  moving  uniformly  forward  in  this 
medium. 

CASE  1.  Let  a  cylinder  described  with  the  same  diameter  and  altitude  be 
conceived  to  go  forward  with  the  same  velocity  in  the  direction  of  its  axis 
through  the  same  medium ;  and  let  us  suppose  that  the  particles  of  the 
medium,  on  which  the  globe  or  cylinder  falls,  fly  back  with  as  great  a  force 
of  reflexion  as  possible.  Then  since  the  resistance  of  the  globe  (by  the  last 
Proposition)  is  but  half  the  resistance  of  the  cylinder,  and  since  the  globe 
is  to  the  cylinder  as  2  to  3,  and  since  the  cylinder  by  falling  perpendicu 
larly  on  the  particles,  and  reflecting  them  with  the  utmost  force,  commu 
nicates  to  them  a  velocity  double  to  its  own;  it  follows  that  the  cylinder. 
in  moving  forward  uniformly  half  the  length  of  its  axis,  will  communicate 
a  motion  to  the  particles  which  is  to  the  whole  motion  of  the  cylinder  as 
the  density  of  the  medium  to  the  density  of  the  cylinder ;  and  that  the 
globe,  in  the  time  it  describes  one  length  of  its  diameter  in  moving  uni 
formly  forward,  will  communicate  the  same  motion  to  the  particles ;  and 
in  the  time  that  it  describes  twro  thirds  of  its  diameter,  will  communicate 
a  motion  to  the  particles  which  is  to  the  whole  motion  of  the  globe  as  the 
density  of  the  medium  to  the  density  of  the  globe.  Arid  therefore  the 
globe  meets  with  a  resistance,  which  is  to  the  force  by  which  its  whole  mo 
tion  may  be  either  taken  away  or  generated  in  the  time  in  which  it  de 
scribes  two  thirds  of  its  diameter  moving  uniformly  forward,  as  the  den 
sity  of  the  medium  to  the  density  of  the  globe. 

CASE  2.  Let  us  suppose  that  the  particles  of  the  medium  incident  on 
the  globe  or  cylinder  are  not  reflected ;  and  then  the  cylinder  falling  per 
pendicularly  on  the  particles  will  communicate  its  own  simple  velocity  to 
them,  and  therefore  meets  a  resistance  but  half  so  great  as  in  the  former 
case,  and  the  globe  also  meets  with  a  resistance  but  half  so  great. 

CASE  3.  Let  us  suppose  the  particles  of  the  medium  to  fly  back  from 
the  globe  with  a  force  which  is  neither  the  greatest,  nor  yet  none  at  all,  but 
with  a  certain  mean  force ;  then  the  resistance  of  the  globe  will  be  in  the 
same  mean  ratio  between  the  resistance  in  the  first  case  and  the  resistance 
in  the  second.  Q.E.I. 

COR.  1.  Hence  if  the  globe  and  the  particles  are  infinitely  hard,  and 
destitute  of  all  elastic  force,  and  therefore  of  all  force  of  reflexion  ;  thf 
resistance  of  the  globe  will  be  to  the  force  by  which  its  whole  motion  may 


330  THE    MATHEMATICAL    PRINCIPLES  [BOOK     I) 

be  destroyed  or  generated,  in  the  time  that  the  globe  describes  four  third 
parts  of  its  diameter,  as  the  density  of  the  medium  to  the  density  of  the 
^lobe. 

Con.  2.  The  resistance  of  the  globe,  cceteris  paribus,  is  in  the  duplicate 
ratio  of  the  velocity. 

CUR.  3.  The  resistance  of  the  globe,  cocteris  paribus,  is  in  the  duplicate 
ratio  of  the  diameter. 

COR.  4.  The  resistance  of  the  globe  is,  cceteris  paribus,  as  the  density  of 
the  medium. 

COR,  5.  The  resistance  of  the  globe  is  in  a  ratio  compounded  of  the  du 
plicate  ratio  of  the  velocity,  arid  the  duplicate  ratio  of  the  diameter,  and 
the  ratio  of  the  density  of  the  medium. 

COR.  6.  The  motion  of  the  globe  and  its  re 
sistance  may  be  thus  expounded  Let  AB  be  the 
time  in  which  the  globe  may,  by  its  resistance 
uniformly  continued,  lose  its  whole  motion. 
Erect  AD,  BC  perpendicular  to  AB.  J  ,et  BC  be 
that  whole  motion,  and  through  the  point  C,  the 
asymptotes  being  AD,  AB,  describe  the  hyperbola 
CF.  Produce  AB  to  any  point  E.  Erect  the  perpendicular  EF  meeting 
the  hyperbola  in  F.  Complete  the  parallelogram  CBEG,  and  draw  AF 
meeting  BC  in  H.  Then  if  the  globe  in  any  time  BE,  with  its  first  mo 
tion  BC  uniformly  continued,  describes  in  a  non-resisting  medium  the  space 
CBEG  expounded  by  the  area  of  the  parallelogram,  the  same  in  a  resisting 
medium  will  describe  the  space  CBEF  expounded  by  the  area  of  the  hv- 
perbola;  and  its  motion  at  the  end  of  that  time  will  be  expounded  by  EF, 
the  ordinate  of  the  hyperbola,  there  being  lost  of  its  motion  the  part  FG. 
And  its  resistance  at  the  end  of  the  same  time  will  be  expounded  by  the 
length  BH,  there  being  lost  of  its  resistance  the  part  CH.  All  these  things 
appear  by  Cor.  1  and  3,  Prop.  V.,  Book  II. 

COR.  7.  Hence  if  the  globe  in  the  time  T  by  the  resistance  R  uniformly 
continued  lose  its  whole  motion  M,  the  same  globe  in  the  time  t  in  a 
resisting  medium,  wherein  the  resistance  R  decreases  in  a  duplicate 

/M 

ratio  of  the  velocity,  will  lose  out  of  its  motion  M  the  part  ,.i '  the 

TM 

part  rn  .  ;  remaining ;  and  will  describe  a  space  which  is  to  the  space  de 
scribed  in  the  same  time  t,  with  the  uniform  motion  M,  as  the  logarithm  of 

T  +  t 
the  number  — ^.—  multiplied  by  the  number  2,302585092994  is  to  the 

number  ^  because  the  hyperbolic  area  BCFE  is  to  the  rectangle  BCGE 
in  that  proportion. 


SEC.   VII.] 


OF    NATURAL    PHILOSOPHY. 


331 


SCHOLIUM. 

I  have  exhibited  in  this  Proposition  the  resistance  and  retardation  of 
spherical  projectiles  in  mediums  that  are  not  continued,  and  shewn  that 
this  resistance  is  to  the  force  by  which  the  whole  motion  of  the  globe  may  be 
destroyed  or  produced  in  the  time  in  which  the  globe  can  describe  two  thirds 
of  its  diameter,  with  a  velocity  uniformly  continued,  as  the  density  of  the 
medium  to  the  density  of  the  globe,  if  so  be  the  globe  and  the  particles  of 
the  medium  be  perfectly  elastic,  and  are  endued  with  the  utmost  force  of 
reflexion ;  and  that  this  force,  where  the  globe  and  particles  of  the  medium 
are  infinitely  hard  and  void  of  any  reflecting  force,  is  diminished  one  half. 
But  in  continued  mediums,  as  water,  hot  oil,  and  quicksilver,  the  globe  as 
it  passes  through  them  does  not  immediately  strike  against  all  the  parti 
cles  of  the  fluid  that  generate  the  resistance  made  to  it,  but  presses  only 
the  particles  that  lie  next  to  it,  which  press  the  particles  beyond,  which 
press  other  particles,  and  so  on  ;  and  in  these  mediums  the  resistance  is  di 
minished  one  other  half.  A  globe  in  these  extremely  fluid  mediums  meets 
with  a  resistance  that  is  to  the  force  by  which  its  whole  motion  may  be 
destroyed  or  generated  in  the  time  wherein  it  can  describe,  with  that  mo 
tion  uniformly  continued,  eight  third  parts  of  its  diameter,  as  the  density 
of  the  medium  to  the  density  of  the  globe.  This  I  shall  endeavour  to  shew 
in  what  follows. 


PROPOSITION  XXXVI.     PROBLEM  VIII. 

To  define  the  motion  of  water  running  out  of  a  cylindrical  vessel  through 

a  hole  made  at  the  bottom. 

Let  AC  D  B  be  a  cylindrical  vessel,  AB  the  mouth  p  =  Q: 

of  it,  CD  the  bottom  p  irallel  to  the  horizon,  EF  a 
circular  hole  in  the  middle  of  the  bottom,  G  the 
c-?ritre  of  the  hole,  and  GH  the  axis  of  the  cylin-  Kj 
cler  perpendicular  to  the  horizon.     And  suppose  a 
cylinder  of  ice  APQ,B  to  be  of  the  same  breadth 
with  the  cavity  of  the  vessel,  and  to  have  the  same 
axis,  and  to  descend  perpetually  with  an  uniform 
motion,  and  that  its  parts,  as  soon  as  they  touch  the 
superficies   AB,   dissolve    into    water,   and    flow 
(  wn  by  their  weight  into  the  vessel,  and  in  their 
fall   compose   the  cataract    or  column    of  water 
ABNFEM,  passing  through  the  hole  EF,  and  filling  up  the  same  exactly. 
Let  the  uniform  velocity  of  the  descending  ice  and  of  the  contiguous  water 
in  the  circle  AB  be  that  which  the  water  would  acquire  by  falling  through 
the  space  IH  ;  and  let  IH  and  HG  lie  in  the  same  right  line ;  and  through 


332  THE    MATHEMATICAL    PRINCIPLES  [BOOK    Jl 

the  point  I  let  there  be  drawn  the  right  line  KL  parallel  to  the  horizon 
and  meeting  the  ice  on  both  the  sides  thereof  in  K  and  L.  Then  the  ve 
locity  of  the  water  running  out  at  the  hole  EF  will  be  the  same  that  it 
would  acquire  by  falling  from  I  through  the  space  IG.  Therefore,  by 
Galih'cJ's  Theorems,  IG  will  be  to  IH  in  the  duplicate  ratio  of  the  velo 
city  of  the  water  that  runs  out  at  the  hole  to  the  velocity  of  the  wrater  in 
the  circle  AB,  that  is,  in  the  duplicate  ratio  of  the  circle  AB  to  the  circle 
EF ;  those  circles  being  reciprocally  as  the  velocities  of  the  water  which 
in  the  same  time  and  in  equal  quantities  passes  severally  through  each  of 
them,  and  completely  fills  them  both.  We  are  now  considering  the  velo 
city  with  which  the  water  tends  to  the  plane  of  the  horizon.  But  the  mo 
tion  parallel  to  the  same,  by  which  the  parts  of  the  falling  water  approach  to 
each  other,  is  not  here  taken  notice  of;  since  it  is  neither  produced  by 
gravity,  nor  at  all  changes  the  motion  perpendicular  to  the  horizon  which  the 
gravity  produces.  We  suppose,  indeed,  that  the  parts  of  the  water  cohere 
a  little,  that  by  their  cohesion  they  may  in  falling  approach  to  each  othei 
with  motions  parallel  to  the  horizon  in  order  to  form  one  single  cataract, 
and  to  prevent  their  being  divided  into  several :  but  the  motion  parallel  to 
the  horizon  arising  from  this  cohesion  does  not  come  under  our  present 
consideration. 

CASE  1.  Conceive  now  the  w^hole  cavity  in  the  vessel,  wrhich  encompasses 
the  falling  water  ABNFEM,  to  be  full  of  ice,  so  that  the  water  may  pass 
through  the  ice  as  through  a  funnel.  Then  if  the  water  pass  very  near  to 
the  ice  only,  without  touching  it;  or,  which  is  the  same  tiling,  if  by  rea 
son  of  the  perfect  smoothness  of  the  surface  of  the  ice,  the  water,  though 
touching  it.  glides  over  it  writh  the  utmost  freedom,  and  without  the  le-ast 
resistance;  the  water  will  run  through  the  hole  EF  with  the  same  velocity 
as  before,  and  the  whole  weight  of  the  column  of  water  ABNFEM  will  be 
all  taken  up  as  before  in  forcing  out  the  water,  and  the  bottom  of  the  vessel 
will  sustain  the  weight  of  the  ice  encompassing  that  column. 

Let  now  the  ice  in  the  vessel  dissolve  into  water ;  yet  will  the  efflux  of 
the  water  remain,  as  to  its  velocity,  the  same  as  before.  It  will  not  be 
less,  because  the  ice  now  dissolved  will  endeavour  to  descend ;  it  will  not 
be  greater,  because  the  ice.  now  become  water,  cannot  descend  without  hin 
dering  the  descent  of  other  water  equal  to  its  own  descent.  The  same  force 
ought  always  to  generate  the  same  velocity  in  the  effluent  water. 

But  the  hole  at  the  bottom  of  the  vessel,  by  reason  of  the  oblique  mo 
tions  of  the  particles  of  the  effluent  water,  must  be  a  little  greater  than  before, 
For  now  the  particles  of  the  water  do  not  all  of  them  pass  through  the 
hole  perpendicularly,  but,  flowing  down  on  all  parts  from  the  sides  of  the 
vessel,  and  converging  towards  the  hole,  pass  through  it  with  oblique  mo 
tions  :  r,r,d  in  tending  downwards  meet  in  a  stream  whose  diameter  is  a  little 
smaller  below  the  hole  than  at  the  hole  itself :  its  diameter  being  to  the 


SEC.  V1L! 


OF    NATURAL    PHILOSOPHY. 


333 


diameter  of  the  hole  as  5  to  6,  or  as  5^  to  6|,  very  nearly,  if  I  took  the 
measures  of  those  diameters  right.  I  procured  a  very  thin  flat  plate,  hav 
ing  a  hole  pierced  in  the  middle,  the  diameter  of  the  circular  hole  being 
f  parts  of  an  inch.  And  that  the  stream  of  running  waters  might  not  be 
accelerated  in  falling,  and  by  that  acceleration  become  narrower,  I  fixed 
this  plate  not  to  the  bottom,  but  to  the  side  of  the  vessel,  so  us  to  make  the 
water  go  out  in  the  direction  of  a  line  parallel  to  the  horizon.  Then,  when 
the  vessel  was  full  of  water,  I  opened  the  hole  to  let  it  run  out ;  and  the 
diameter  of  the  stream,  measured  with  great  accuracy  at  the  distance  of 
about  half  an  inch  from  the  hole,  was  f  J-  of  an  inch.  Therefore  the  di 
ameter  of  this  circular  hole  was  to  the  diameter  of  the  stream  very  nearly 
as  25  to  21.  So  that  the  water  in  passing  through  the  hole  converges  on 
all  sides,  and,  after  it  has  run  out  of  the  vessel,  becomes  smaller  by  converg 
ing  in  that  manner,  and  by  becoming  smaller  is  accelerated  till  it  comes  to 
the  distance  of  half  an  inch  from  the  hole,  and  at  that  distance  flows  in  a 
smaller  stream  and  with  greater  celerity  than  in  the  hole  itself,  and  this 
in  the  ratio  of  25  X  25  to  21  X  21,  or  17  to  12,  very  nearly ;  that  is,  in 
about  the  subdaplicate  ratio  of  2  to  1.  Now  it  is  certain  from  experiments, 
that  the  quantity  of  water  running  out  in  a  given  time  through  a  circular 
hole  made  in  the  bottom  of  a  vessel  is  equal  to  the  quantity,  which,  flow 
ing  with  the  aforesaid  velocity,  would  run  out  in  the  same  time  through 
another  circular  hole,  whose  diameter  is  to  the  diameter  of  the  former  as 
21  to  25.  And  therefore  that  running  water  in  passing  through  the 
hole  itself  has  a  velocity  downwards  equal  to  that  which  a  heavy  body 
would  acquire  in  falling  through  half  the  height  of  the  stagnant  water  in 
the  vessel,  nearly.  But,  then,  after  it  has  run  out,  it  is  still  accelerated  by 
converging,  till  it  arrives  at  a  distance  from  the  hole  that  is  nearly  equal  to 
its  diameter,  and  acquires  a  velocity  greater  than  the  other  in  about  the 
subduplicate  ratio  of  2  to  1 ;  which  velocity  a  heavy  body  would  nearly 
acquire  by  falling  through  the  whole  height  of  the  stagnant  water  in  the 
vessel. 

Therefore  in  what  follows  let  the  diameter  of 
the  stream  be  represented  by  that  lesser  hole  which 
we  called  EF.  And  imagine  another  plane  VW 
above  the  hole  EF,  and  parallel  to  the  plane  there 
of,  to  be  placed  at  a  distance  equal  to  the  diame 
ter  of  the  same  hole,  and  to  be  pierced  through 
with  a  greater  hole  ST,  of  such  a  magnitude  that 
a  stream  which  will  exactly  fill  the  lower  hole  EF 
may  pass  through  it ;  the  diameter  of  which  hole 
will  therefore  be  to  the  diameter  of  the  lower  hole  as  25  to  21,  nearly.  By 
this  means  the  water  will  run  perpendicularly  out  at  the  lower  hole ;  and 
the  quantity  of  the  water  running  out  will  be,  according  to  the  magnitude 


334  THE    MATHEMATICAL    PRINCIPLES  [BOOK    11 

of  this  last  hole,  the  same,  very  nearly,  which  the  solution  of  the  Problem 
requires.  The  space  included  between  the  two  planes  and  the  falling  stream 
may  be  considered  as  the  bottom  of  the  vessel.  But,  to  make  the  solution 
more  simple  and  mathematical,  it  is  better  to  take  the  lower  plane  alone 
for  the  bottom  of  the  vessel,  and  to  suppose  that  the  water  which  flowed 
through  the  ice  as  through  a  funnel,  and  ran  out  of  the  vessel  through  the 
hole  EF  made  in  the  lower  plane,  preserves  its  motion  continually,  and  that 
the  ice  continues  at  rest.  Therefore  in  what  follows  let  ST  be  the  diame 
ter  of  a  circular  hole  described  from